Download Helping You Remember

Teacher Annotated Edition
Study Notebook
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted
under the United States Copyright Act, no part of this publication may be reproduced or
distributed in any form or by any means, or stored in a database or retrieval system, without
prior written permission of the publisher.
Send all inquiries to:
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, OH 43240
ISBN: 978-0-07-890858-3
MHID: 0-07-890858-2
Printed in the United States of America
1 2 3 4 5 6 7 8 9 10 009 14 13 12 11 10 09 08
Contents
Chapter 1
Chapter 5
Before You Read ............................................. 1
Key Points ........................................................ 2
1-1 Points, Lines, and Planes ........................... 3
1-2 Linear Measure and Precision ................... 5
1-3 Distance and Midpoints .............................. 7
1-4 Angle Measure ........................................... 9
1-5 Angle Relationships .................................. 11
1-6 Two-Dimensional Figures ......................... 13
1-7 Three-Dimensional Figures ...................... 15
Tie It Together................................................ 17
Before the Test .............................................. 18
Before You Read ........................................... 75
Key Points ...................................................... 76
5-1 Bisectors of Triangles ............................... 77
5-2 Medians and Altitudes of Triangles .......... 79
5-3 Inequalities in One Triangle ..................... 81
5-4 Indirect Proof ............................................ 83
5-5 The Triangle Inequality ............................. 85
5-6 Inequalities in Two Triangles .................... 87
Tie It Together................................................ 89
Before the Test .............................................. 90
Chapter 2
Before You Read ........................................... 91
Key Points ...................................................... 92
6-1 Angles of Polygons................................... 93
6-2 Parallelograms.......................................... 95
6-3 Tests for Parallelograms .......................... 97
6-4 Rectangles................................................ 99
6-5 Rhombi and Squares.............................. 101
6-6 Kites and Trapezoids.............................. 103
Tie It Together.............................................. 105
Before the Test ............................................ 106
Chapter 6
Before You Read ........................................... 19
Key Points ...................................................... 20
2-1 Inductive Reasoning and Conjecture ....... 21
2-2 Logic ......................................................... 23
2-3 Conditional Statements ............................ 25
2-4 Deductive Reasoning ............................... 27
2-5 Postulates and Paragraph Proofs ............ 29
2-6 Algebraic Proof ......................................... 31
2-7 Proving Segment Relationships ............... 33
2-8 Proving Angle Relationships .................... 35
Tie It Together................................................ 37
Before the Test .............................................. 38
Chapter 7
Before You Read ......................................... 107
Key Points .................................................... 108
7-1 Ratios and Proportions ........................... 109
7-2 Similar Polygons ..................................... 111
7-3 Similar Triangles ..................................... 113
7-4 Parallel Lines and Proportional
Parts ....................................................... 115
7-5 Parts of Similar Triangles ....................... 117
7-6 Similarity Transformations ...................... 119
7-7 Scale Drawings and Models ................... 121
Tie It Together.............................................. 123
Before the Test ............................................ 124
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 3
Before You Read ........................................... 39
Key Points ...................................................... 40
3-1 Parallel Lines and Transversals ............... 41
3-2 Angles and Parallel Lines ......................... 43
3-3 Slopes of Lines ......................................... 45
3-4 Equations of Lines .................................... 47
3-5 Proving Lines Parallel............................... 49
3-6 Perpendiculars and Distance ................... 51
Tie It Together................................................ 53
Before the Test .............................................. 54
Chapter 8
Chapter 4
Before You Read ......................................... 125
Key Points .................................................... 126
8-1 Geometric Mean ..................................... 127
8-2 The Pythagorean Theorem and Its
Converse ................................................ 129
8-3 Special Right Triangles .......................... 131
8-4 Trigonometry .......................................... 133
8-5 Angles of Elevation and Depression ...... 135
8-6 The Law of Sines and Cosines .............. 137
8-7 Vectors ................................................... 139
Tie It Together.............................................. 141
Before the Test ............................................ 142
Before You Read ........................................... 55
Key Points ...................................................... 56
4-1 Classifying Triangles ................................ 57
4-2 Angles of Triangles................................... 59
4-3 Congruent Triangles ................................. 61
4-4 Proving Congruence: SSS, SAS .............. 63
4-5 Proving Congruence: ASA, AAS .............. 65
4-6 Isosceles and Equilateral Triangles ......... 67
4-7 Congruence Transformations ................... 69
4-8 Triangles and Coordinate Proof ............... 71
Tie It Together................................................ 73
Before the Test .............................................. 74
iii
Chapter 9
Before You Read ......................................... 143
Key Points .................................................... 144
9-1 Reflections .............................................. 145
9-2 Translations ............................................ 147
9-3 Rotations ................................................ 149
9-4 Compositions of Transformations........... 151
9-5 Symmetry ............................................... 153
9-6 Dilations .................................................. 155
Tie It Together.............................................. 157
Before the Test ............................................ 158
Chapter 10
Before You Read ......................................... 159
Key Points .................................................... 160
10-1 Circles and Circumference ................... 161
10-2 Measuring Angles and Arcs ................. 163
10-3 Arcs and Chords................................... 165
10-4 Inscribed Angles ................................... 167
10-5 Tangents............................................... 169
10-6 Secants, Tangents, and Angle
Measures .............................................. 171
10-7 Special Segments in a Circle ............... 173
10-8 Equations of Circles ............................. 175
Tie It Together.............................................. 177
Before the Test ............................................ 178
Chapter 11
Before You Read ......................................... 193
Key Points .................................................... 194
12-1 Representations of
Three-Dimensional Figures .................. 195
12-2 Surface Areas of Prisms and
Cylinders............................................... 197
12-3 Surface Areas of Pyramids and
Cones ................................................... 199
12-4 Volumes of Prisms and Cylinders ........ 201
12-5 Volumes of Pyramids and Cones ......... 203
12-6 Surface Areas and Volumes
of Spheres ............................................ 205
12-7 Spherical Geometry .............................. 207
12-8 Congruent and Similar Solids ............... 209
Tie It Together.............................................. 211
Before the Test ............................................ 212
Chapter 13
Before You Read ......................................... 213
Key Points .................................................... 214
13-1 Representing Sample Spaces .............. 215
13-2 Permutations and Combinations .......... 217
13-3 Geometric Probability ........................... 219
13-4 Simulations ........................................... 221
13-5 Probabilities of Independent and
Dependent Events ................................ 223
13-6 Probabilities of Mutually Exclusive
Events................................................... 225
Tie It Together.............................................. 227
Before the Test ............................................ 228
iv
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Before You Read ......................................... 179
Key Points .................................................... 180
11-1 Areas of Parallelograms and
Triangles ............................................... 181
11-2 Areas of Trapezoids, Rhombi
and Kites .............................................. 183
11-3 Areas of Circles and Sectors................ 185
11-4 Areas of Regular Polygons and
Composite Figures ............................... 187
11-5 Areas of Similar Figures ....................... 189
Tie It Together.............................................. 191
Before the Test ............................................ 192
Chapter 12
Note-Taking Tips
Your notes are a reminder of what you learned in class. Taking good notes can help you
succeed in mathematics. The following tips will help you take better classroom notes.
• Before class, ask what your teacher will be discussing in class. Review mentally what you
already know about the concept.
• Be an active listener. Focus on what your teacher is saying. Listen for important
concepts. Pay attention to words, examples, and/or diagrams your teacher emphasizes.
• Write your notes as clear and concise as possible. The following symbols and
abbreviations may be helpful in your note-taking.
Word or
Phrase
Symbol or
Abbreviation
for example
e.g.
not equal
≠
such as
i.e.
approximately
≈
with
w/
therefore
∴
without
w/o
versus
vs
+
angle
∠
and
Word or
Phrase
Symbol or
Abbreviation
• Use a symbol such as a star (%) or an asterisk (∗) to emphasis important concepts. Place
a question mark (?) next to anything that you do not understand.
• Ask questions and participate in class discussion.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
• Draw and label pictures or diagrams to help clarify a concept.
• When working out an example, write what you are doing to solve the problem next to
each step. Be sure to use your own words.
• Review your notes as soon as possible after class. During this time, organize and
summarize new concepts and clarify misunderstandings.
Note-Taking Don’ts
• Don’t write every word. Concentrate on the main ideas and concepts.
• Don’t use someone else’s notes as they may not make sense.
• Don’t doodle. It distracts you from listening actively.
• Don’t lose focus or you will become lost in your note-taking.
v
NAME
DATE
PERIOD
The Tools of Geometry
Before You Read
Before you read the chapter, respond to these statements.
1. Write an A if you agree with the statement.
2. Write a D if you disagree with the statement.
Before You Read
The Tools of Geometry
See students’ work.
• Collinear points are lines that run
through the same point.
• Line segments that are congruent have
the same measure.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
• Distance on a coordinate plane is
calculated using a form of the
Pythagorean Theorem.
• The Midpoint Formula can be used to
find the coordinates of the endpoint of a
segment.
• The formula to find the area of a circle
is A = πr2.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• When taking notes, summarize the main ideas presented in the lesson.
Summaries are useful for condensing data and realizing what is important.
• When you take notes, write descriptive paragraphs about your learning
experiences.
Chapter 1
1
Glencoe Geometry
NAME
DATE
PERIOD
The Tools of Geometry
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on linear measure, one fact might be that unlike a line, a line
segment can be measured because it has two endpoints. After completing the chapter, you
can use this table to review for your chapter test.
Lesson
Fact
See students’ work.
1-1 Points, Lines, and Planes
1-2 Linear Measure
1-3 Distance and Midpoints
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1-4 Angle Measure
1-5 Angle Relationship
1-6 Two-Dimensional Figures
1-7 Three-Dimensional Figures
Chapter 1
2
Glencoe Geometry
NAME
1-1
DATE
PERIOD
Points, Lines, and Planes
What You’ll Learn
Scan the text in Lesson 1-1. Write two facts you learned
about points, lines, and planes as you scanned the text.
1.
Sample
answer: A line does not have thickness or
____________________________________________________
2.
Sample
answer: Collinear points are points on the
____________________________________________________
same
line.
____________________________________________________
Active Vocabulary
undefined term
point
line
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
plane
New Vocabulary Write the definition next to each term.
can only be explained using examples and descriptions
————————————————————————————
a location that has neither shape nor size
————————————————————————————
made up of points and has no thickness or width
————————————————————————————
flat surface made up of points that extends
————————————————————————————
indefinitely in all directions
————————————————————————————
collinear
points that lie on the same line
coplanar
points that lie on the same plane
intersection
————————————————————————————
————————————————————————————
the set of points that two or more geometric figures
————————————————————————————
have in common
————————————————————————————
definition/defined term
can be explained using undefined terms and/or other
————————————————————————————
defined terms
————————————————————————————
space
Chapter 1
a boundless, three-dimensional set of all points
————————————————————————————
3
Glencoe Geometry
Lesson 1-1
width.
____________________________________________________
NAME
DATE
PERIOD
Lesson 1-1 (continued)
Main Idea
Points, Lines, and
Planes
pp. 5–6
Details
Model a point, line, and plane with a representative
drawing. Label your drawing.
Sample answers:
•
point
line
plane
Intersections of Lines
and Planes
pp. 6–7
Compare undefined and defined terms by completing
the table. Provide definitions and examples.
Definition
Examples
defined term
explained using
undefined terms
or other defined
terms; same as
definition
space,
intersection,
collinear,
coplanar,
intersection
can only be
explained using
undefined term
examples and
descriptions
points, lines,
planes
Helping You Remember
Recall or look in a dictionary for the meaning of
the prefix co-. What does the prefix mean? How can it help you remember the meaning
of collinear?
Sample answer: joint or working together; two or more points together
—————————————————————————————————————————
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 1
4
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Term
NAME
DATE
1-2
Linear Measure
Active Vocabulary
betweenness of points
line segment
between
Scan the text under the Now heading. List two things you
will learn about in this lesson.
1.
Sample answer: how to measure segments
2.
Sample answer: how to calculate with measures
New Vocabulary Fill in each blank with the correct term or
phrase.
Between any two
point
points
on a line there is another
.
line
a portion of a
with two
endpoints
point A is between two other points, B and C; A, B, and C
are on the same line, and AB + AC = BC
construction
method of creating geometric figures without
measuring tools
congruent segment
two
segments with the same measures
Vocabulary Link Congruent segments can be illustrated by
real-world examples. Consider opposite sides of this book.
The lines are the same size. Write some real-world examples
of other congruent segments.
Sample answer: railroad tracks, edges of picture
————————————————————————————
frame
————————————————————————————
————————————————————————————
Chapter 1
5
Glencoe Geometry
Lesson 1-2
What You’ll Learn
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
PERIOD
NAME
DATE
PERIOD
Lesson 1-2 (continued)
Main Idea
Measure Line Segments
pp. 14–15
Details
Use the model to fill in each blanks.
A
B
JO
Calculate Measures
pp. 15–17
1.
eighths
Each inch is divided into _________.
2.
3
1−
8
Point B is closer to the ______
inch mark.
3.
3
1−
−−
8
AB is about ______
inches long.
Model the three types of methods to calculate
measures of a line segment with an example.
Find
measurements
by subtracting.
Sample
answer:
number line
with two
segments that
are equal to a
longer
segment
Sample
answer:
number line
with a smaller
segment
subtracted
from a larger
segment
Find
measurements by
writing and
solving equations.
Sample answer:
ST = 4x,
SV = 5x - 3,
TV = 15
Find the value of
x and ST.
4x + 5x - 3 = 15
x = 2; ST = 8
Helping You Remember
Construction is a word used in everyday English
as well as in mathematics. Look up construction in the dictionary. Explain how the
everyday definition can help you remember how construction is used in mathematics.
Sample answer: The way in which something is put together; figures are put
—————————————————————————————————————————
together just like buildings and structures.
—————————————————————————————————————————
Chapter 1
6
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find
measurements
by adding.
NAME
1-3
DATE
PERIOD
Distance and Midpoints
What You’ll Learn
Skim Lesson 1-3. Predict two things that you expect to learn
based on the headings and the Key Concept box.
1.
Sample
answer: how to find the distance between
____________________________________________________
two
points
____________________________________________________
2.
Sample
answer: how to find the midpoint between
____________________________________________________
two
points
____________________________________________________
Active Vocabulary
distance
New Vocabulary Write the definition next to each term.
the length of a segment with two endpoints
————————————————————————————
————————————————————————————
point that is halfway between the endpoints
————————————————————————————
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
————————————————————————————
segment bisector
segment, line, plane that intersects a segment at
————————————————————————————
its midpoint
————————————————————————————
Vocabulary Link Midpoint can have non-mathematical
meanings as well. Look up midpoint in the dictionary.
Explain how the English definition can help you remember
how midpoint is used in mathematics.
Sample answer: a point in time halfway between
————————————————————————————
the beginning and end; a point halfway between
————————————————————————————
two extremes
————————————————————————————
————————————————————————————
Chapter 1
7
Glencoe Geometry
Lesson 1-3
midpoint
NAME
DATE
PERIOD
Lesson 1-3 (continued)
Main Idea
Details
Distance Between Two
Points
pp. 25–26
Fill in the organizer for the distance formula on a
number line and a coordinate plane with line segment
AB, with endpoints (x1, y1) and (x2, y2).
Distance Formula
Number Line
Coordinate Plane
(x2 - x1)2 + (y2 - y1)2
√"""""""""
If a segment has
endpoints with
coordinates (x1, y1)
and (x2, y2), you can
find the distance
with the distance
formula.
Words
Model the solution to find the distance between the
points on the coordinate plane. Use the lines to show
your calculations. Round to the nearest tenth if
necessary.
(x1, y1) = (-5, 3) ; (x2, y2) = ( 2, 4 )
KL =
(x - x ) + (y - y )
√""""""""""
KL =
(2 - (-5))2 + (4 - 3)2
√"""""""""""
2
1
2
KL = √"""
49 + 1
2
1
y
2
-
,
0
x
KL = √""
50 ≈ 7.1
Chapter 1
8
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The distance
between two
points is the
absolute value of
the difference
between their
coordinates.
Midpoint of a Segment
pp. 27–30
AB =
Symbols
AB = x1 - x2 or x2 - x1
NAME
1-4
DATE
PERIOD
Angle Measure
What You’ll Learn
Skim the Examples for Lesson 1-4. Predict two things you
think you will learn about angle measures.
1.
Sample
answer: how to measure angles
____________________________________________________
____________________________________________________
2.
Sample
answer: how to classify angles
____________________________________________________
____________________________________________________
right angle
vertex
obtuse angle
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
angle bisector
interior
angle with measure of 90º
where two rays meet to form an angle
angle with measure greater than 90º
a ray that divides an angle into two congruent angles
the region of a plane inside of an angle
acute angle
angle with measure less than 90º
side
a ray that forms part of an angle
ray
angle
degree
opposite rays
exterior
Chapter 1
New Vocabulary Write the correct term next to each
definition.
part of a line with one endpoint and one end that extends
forever
formed by two noncollinear rays with a common endpoint
units used to measure angles
extend in two directions from a point on a line
the region of a plane outside of an angle
9
Glencoe Geometry
Lesson 1-4
Active Vocabulary
NAME
DATE
PERIOD
Lesson 1-4 (continued)
Main Idea
Measure and Classify
Angles
pp. 36–38
Details
Summarize information about angles in the graphic
organizer below.
1.
Draw and label an obtuse, acute, and right angle.
obtuse
acute
right
2.
Draw and label one angle that has these 3 names:
∠B, ∠ABC, and ∠3.
"
#
Congruent Angles
pp. 39–40
3
$
Model a pair of congruent angles. Use a compass and
straightedge.
Drawings will vary, but both angles have the same
measure. Sample answer shown.
Compare and contrast congruent segments and
congruent angles.
Sample answer: Congruent segments and congruent angles have the
—————————————————————————————————————————
same measure. Segments have a linear measure. Angles are measured
—————————————————————————————————————————
in degrees.
—————————————————————————————————————————
Chapter 1
10
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Helping You Remember
NAME
1-5
DATE
PERIOD
Angle Relationships
What You’ll Learn
Scan Lesson 1-5. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: Special Pairs of Lines
____________________________________________________
____________________________________________________
2.
Sample
answer: Perpendicular Lines
____________________________________________________
____________________________________________________
Active Vocabulary
New Vocabulary Label the diagram with the correct terms.
Use each term once.
adjacent
linear pair
vertical
complementary
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
supplementary
perpendicular
∠1 and ∠2 are
supplementary angles
and a linear pair.
w
Lines w and x are
perpendicular.
1 5
4
∠4 and ∠5 are
complementary angles
and adjacent angles.
2 3
∠1 and ∠3 are
vertical angles.
x
Vocabulary Link Perpendicular has a nonmathematical
meaning. Look up perpendicular in the dictionary. List at
least five examples of things that are perpendicular in
everyday life.
Sample answer: lines of corner of book, desk, room,
refrigerator and floor, wall and ceiling
————————————————————————————
————————————————————————————
Chapter 1
11
Glencoe Geometry
Lesson 1-5
————————————————————————————
NAME
DATE
PERIOD
Lesson 1-5 (continued)
Main Idea
Pairs of Angles
pp. 46–48
Details
Summarize angle relationships by completing the
graphic organizer with the terms adjacent angles,
complementary angles, vertical angles, and
supplementary angles.
adjacent
angles
vertical
angles
relationship due
to positioning
Special angle pairs
relationship due
to angle measures
complementary
angles
# ⊥ BA
#.
If m∠ABC = 2x + 12, find x so that BC
"
&
%
#
$
2x + 12
=
90
definition of perpendicular
x
=
39
Simplify.
Helping You Remember
Look up the nonmathematical meaning of
supplementary in a dictionary. How can this definition help you to remember the
meaning of supplementary angles?
Sample answer: something added to complete a deficiency; two angles added
—————————————————————————————————————————
together to be a complete line
—————————————————————————————————————————
Chapter 1
12
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Perpendicular Lines
pp. 48–50
supplementary
angles
NAME
DATE
1-6
PERIOD
Two-Dimensional Figures
What You’ll Learn
Skim the lesson. Write two things you already know about
two-dimensional figures.
1.
Sample
answer: Polygons are two-dimensional
____________________________________________________
2.
Sample
answer: Perimeter and area of
____________________________________________________
two-dimensional
figures can be found.
____________________________________________________
Active Vocabulary
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
vertex of polygon
where two sides of a polygon intersect
convex
polygon with no side that passes through the figure’s
interior
regular polygon
a convex polygon that is equilateral and equiangular
equiangular polygon
circumference
n-gon
a polygon where all angles are congruent
perimeter of a circle
a polygon with n number of sides
polygon
closed figure formed by coplanar segments where two
segments intersect at a common noncollinear endpoint
concave
polygon with some side that passes through the figure’s
interior
equilateral polygon
perimeter
area
Chapter 1
New Vocabulary Write each term next to its definition.
a polygon where all sides are congruent
sum of the lengths of a polygon’s sides
number of square units needed to cover a surface
13
Glencoe Geometry
Lesson 1-6
figures.
____________________________________________________
NAME
DATE
PERIOD
Lesson 1-6 (continued)
Main Idea
Identify Polygons
pp. 56–57
Details
Fill in the organizer about polygons.
What is a polygon?
What is not a polygon?
• noncollinear vertices
• closed figure
• made of coplanar
segments
• open
• curved sides
• intersecting segments
Polygons
Examples
Summarize the formulas for perimeter,
circumference, and area of the figures by completing
the chart.
Area
Perimeter
Circumference
Chapter 1
1
A=−
bh
2
A = s2
A =ℓ × w
πr2
same as
P = a + b + c P = 4s P = 2ℓ + 2w circumference
n/a
14
n/a
n/a
C =πd or
C = 2πr
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Perimeter,
Circumference,
and Area
pp. 58–60
Nonexamples
NAME
1-7
DATE
PERIOD
Three-Dimensional Figures
What You’ll Learn
Scan the text in Lesson 1-7. Write two facts you learned
about three-dimensional figures.
1.
Sample
answer: A polyhedron is a solid with all
____________________________________________________
flat
surfaces that enclose a single region or space.
____________________________________________________
____________________________________________________
2.
Sample
answer: There are exactly five types of
____________________________________________________
regular
polyhedron.
____________________________________________________
Active Vocabulary
prism
New Vocabulary Write the definition next to each term.
polyhedron with two bases connected by
————————————————————————————
parallelogram faces
————————————————————————————
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
pyramid
polyhedron with a polygonal base and three or more
————————————————————————————
triangular faces that meet at a common vertex
————————————————————————————
cylinder
solid with congruent parallel circular bases and
————————————————————————————
curved side
————————————————————————————
base
parallel, congruent faces that intersect faces
————————————————————————————
————————————————————————————
cone
solid with a circular base connected by a curved
————————————————————————————
surface to a single vertex
————————————————————————————
sphere
set of points equidistant from a given point
————————————————————————————
————————————————————————————
Chapter 1
15
Glencoe Geometry
Lesson 1-7
____________________________________________________
NAME
DATE
PERIOD
Lesson 1-7 (continued)
Main Idea
Details
Complete the concept circle by drawing in a fourth
example. Then state the relationship of the 4 figures
on the line below the concept circle.
Identify
Three-Dimensional
Figures
pp. 67–68
Drawings
will vary;
must be a
prism.
The relationship is: All are prisms.
————————————————————————————
Organize information about volume and surface area
in the chart.
Surface Area and
Volume
pp. 69–70
Figure
1
T=−
Pℓ T = 2πrh T = πrℓ
T=
2
2
+
πr
4πr 2
+
2πr2
+B
Volume V = Bh
1
V=−
bh
3
1
3
V = πr h V = −πr 2h
2
4
V =−
3
πr3
Helping You Remember
A good way to remember the characteristics of
geometric solids is to think of how different solids are alike. Name a way which
pyramids and cones are alike.
Sample answer: Pyramids and cones both have one common vertex in which
—————————————————————————————————————————
the sides or faces are attached.
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 1
16
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Surface T = Ph +
Area
2B
NAME
DATE
PERIOD
The Tools of Geometry
Tie It Together
Complete the graphic organizer with a term or formula from the chapter.
Vertical
Linear Pair
Adjacent
Length/Distance Formula
d=
Complementary
Angle Pairs
Supplementary
Segment
(x2 - x1) + (y2 - y1)
√#########
Right
Angle
Midpoint Formula
(x1 + x2) (y1 + y2)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
(
)
−, −
2
2
Line
Point
Plane
Obtuse
Perpendicular
Acute
Basic Geometry
Cylinder
Perimeter
Area
3D Figures
Polygons
Equilateral
Equiangular
Volume
Cone
Sphere
Polyhedron
Prism
Surface Area
Regular
Chapter 1
Pyramid
17
Glencoe Geometry
NAME
DATE
PERIOD
The Tools of Geometry
Before the Test
Now that you have read and worked through the chapter, think about what you have
learned and complete the table below. Compare your previous answers with these.
1. Write an A if you agree with the statement.
2. Write a D if you disagree with the statement.
The Tools of Geometry
After You Read
• Collinear points are lines that run through the
same point.
D
• Line segments that are congruent have the same
measure.
A
• Distance on a coordinate plane is calculated using a
form of the Pythagorean Theorem.
• The Midpoint Formula can be used to find the
coordinates of the endpoint of a segment.
• The formula to find the area of a circle is A = πr2.
A
A
A
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 1 Study Guide and Review in the textbook.
□ I took the Chapter 1 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• Make a calendar that includes all of your daily classes. Besides writing down all
assignments and due dates, include in your daily schedule time to study, work on
projects, and reviewing notes you took during class that day.
Chapter 1
18
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 1.
NAME
DATE
PERIOD
Reasoning and Proof
Before You Read
Before you read the chapter, respond to these statements.
1. Write an A if you agree with the statement.
2. Write a D if you disagree with the statement.
Before You Read
Reasoning and Proof
See students’ work.
• Listing a counterexample is a method
to prove a conjecture is true.
• A Venn diagram can illustrate a
conjunction.
• A form of inductive reasoning is the
Law of Detachment.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
• The statement a line contains two
points, is an example of a postulate.
• Supplementary angles have measures
whose sum is 90º.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• When you take notes, you may wish to use a highlighting marker to emphasize
important concepts.
• When you take notes, think about the order in which the concepts are being
presented.
Write why you think the concepts were presented in this sequence.
Chapter 2
19
Glencoe Geometry
NAME
DATE
PERIOD
The Reasoning and Proof
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on inductive reasoning and conjecture, one fact might be a
concluding statement reached using inductive reasoning is called a conjecture. After
completing the chapter, you can use this table to review for your chapter test.
Lesson
2-1 Inductive Reasoning and Conjecture
Fact
See students’ work.
2-2 Logic
2-3 Conditional Statements
2-4 Deductive Reasoning
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2-5 Postulates and Paragraph Proofs
2-6 Algebraic Proof
2-7 Proving Segment Relationships
2-8 Proving Angle Relationships
Chapter 2
20
Glencoe Geometry
NAME
DATE
2-1
PERIOD
Inductive Reasoning and Conjecture
What You’ll Learn
Scan Lesson 2-1. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: Making Conjectures
____________________________________________________
2.
Sample
answer: Finding Counterexamples
____________________________________________________
____________________________________________________
Active Vocabulary
inductive reasoning
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
conjecture
counterexample
New Vocabulary Fill in each blank with the correct term or
phrase.
reasoning that uses a number of specific ___________
examples to
____________
conclusion
arrive at a _____________
conclusion that is reached within inductive reasoning
the _____________
disproves a ____________
conjecture
an example that ____________
Vocabulary Link Conjecture is a word that is used in everyday
English. Find the definition of conjecture using a dictionary.
Write how the definition of conjecture can help you
remember the mathematical definition of conjecture.
Sample answer: an expression of an opinion;
————————————————————————————
A conclusion about a statement is also an expression
————————————————————————————
of an opinion.
————————————————————————————
Chapter 2
21
Glencoe Geometry
Lesson 2-1
____________________________________________________
NAME
DATE
PERIOD
Lesson 2-1 (continued)
Main Idea
Make Conjectures
pp. 89–91
Details
Sequence the steps of making a conjecture with
algebraic terms or geometric terms.
Look for operations or
attributes that repeat.
Ensure all consecutive
terms follow the
same repeating
operation or attribute.
Find Counterexamples
p. 92
Test the conjecture.
Make a conjecture.
Write a statement in which you can make a true
conjecture. Then write another statement in which
you can make a false conjecture. Provide a
counterexample of the false conjecture.
True Conjecture: Sample answer: The product of two
False Conjecture: Sample answer: The product of a
negative number and a positive number is a positive
number. Sample answer: -4 ! 5 = -20
Helping You Remember
Write a short sentence that can help you
remember why it only takes one counterexample to prove that a conjecture is false.
Sample answer: A conjecture must be true in all cases, so one counterexample
—————————————————————————————————————————
is enough to disprove a conjecture.
—————————————————————————————————————————
Chapter 2
22
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
odd numbers is an even number.
NAME
DATE
2-2
PERIOD
Logic
What You’ll Learn
Skim the Examples for Lesson 2-2. Predict two things you
think you will learn about logic.
1.
Sample
answer: how to identify the truth of a
____________________________________________________
statement
____________________________________________________
2.
Sample
answer: how to use Venn diagrams
____________________________________________________
____________________________________________________
truth table
a sentence that is either true or false
disjunction
two or more statements joined by and or or
statement
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
New Vocabulary Match the term with the correct definition
by drawing a line between the two.
Lesson 2-2
Active Vocabulary
a compound statement that uses the word or
truth value
the value of a statement as either true or false
conjunction
statement with the opposite meaning and opposite truth
negation
compound statement
a compound statement that uses the word and
convenient method to determine the truth value of
statement
Vocabulary Link Negation is a word used in everyday English
as well as in mathematics. Look up negation in the
dictionary. Explain how the English definition can help you
remember how negation is used in mathematics.
Sample answer: denying; Negation denies the truth of
————————————————————————————
a statement.
————————————————————————————
Chapter 2
23
Glencoe Geometry
NAME
DATE
PERIOD
Lesson 2-2 (continued)
Main Idea
Details
Determine Truth Values
pp. 97–99
Fill in the blanks to summarize negations,
conjunctions, and disjunctions. Write a description of
each term.
truth value
____________
Statement
compound
statement
disjunction
____________
Venn Diagrams
pp. 99–100
negation
__________
conjunction
____________
Model the situation using a Venn diagram.
At Terrace Middle school, 68 students play basketball, 77
play volleyball, 19 play soccer and basketball, and 27 play
all three sports. If 13 students play both volleyball and
basketball, how many students play just basketball? 9
Basketball
9
27
Volleyball
19
Soccer
Helping You Remember
Prefixes can often help you remember the
meaning of words or distinguish between similar words. Use your dictionary to find the
meanings of the prefixes con and dis and explain how these meanings can help you
remember the difference between a conjunction and disjunction.
Sample answer: Con means together, and dis means undo or do the
————————————————————————————————————————
opposite. Together means and and opposite means or.
————————————————————————————————————————
Chapter 2
24
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
13
NAME
DATE
2-3
PERIOD
Conditional Statements
What You’ll Learn
Skim Lesson 2-3. Predict two things you will learn based on
the headings and the Key Concept box.
1.
Sample
answer: definition of a conditional
____________________________________________________
statement
____________________________________________________
2.
Sample
answer: definition of hypothesis and
____________________________________________________
conclusion
____________________________________________________
Active Vocabulary
conditional statement
if-then statement
converse
New Vocabulary Write the definition next to each term.
a statement that can be written in if-then form
————————————————————————————
in the form if p, then q
————————————————————————————
formed by exchanging the hypothesis and conclusion
————————————————————————————
of the conditional
hypothesis
the phrase immediately following the word if in a
————————————————————————————
conditional statement
————————————————————————————
inverse
formed by negating both the hypothesis and
————————————————————————————
conclusion of the conditional
————————————————————————————
contrapositive
formed by negating both the hypothesis and
————————————————————————————
conclusion of the converse of the conditional
————————————————————————————
logically equivalent
related conditionals
statements with the same truth value
————————————————————————————
other statements based on a given conditional
————————————————————————————
statement
————————————————————————————
conclusion
the phrase immediately following the word then in a
————————————————————————————
conditional statement
————————————————————————————
Chapter 2
25
Glencoe Geometry
Lesson 2-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
————————————————————————————
NAME
DATE
PERIOD
Lesson 2-3 (continued)
Main Idea
If-Then Statements
pp. 105–107
Details
Identify the hypothesis and conclusion of each
conditional statement. Circle the hypothesis and
underline the conclusion.
1. A number is divisible by 9 if the sum of its digits is
divisible by 9.
2. If the measure of an angle is less than 90 degrees, it is an
acute angle.
Related Conditionals
pp. 107–108
Fill in the organizer for related conditionals.
Symbols: p → q
conditional
Words: ______________
statement; if p then q
______________________
Symbols: -p → -q
inverse;
Words: ___________
if not p then not q
___________________
Symbols: q → p
converse;
Words: ___________
if q then p
___________
Symbols: -q → -p
contrapositive;
Words: ________________
if
not q then not p
___________________
Helping You Remember
When working with a conditional statement and
its three related conditional, what is an easy way to remember which statements are
logically equivalent?
Sample answer: The two terms that contain verse (converse and inverse) are
—————————————————————————————————————————
logically equivalent. The other two terms are also logically equivalent.
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 2
26
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Conditional
Statements
NAME
2-4
DATE
PERIOD
Deductive Reasoning
What You’ll Learn
Scan the text in Lesson 2-4. Write two facts you learned
about deductive reasoning as you scanned the text.
1.
Sample
answer: Deductive reasoning uses facts,
____________________________________________________
rules,
definitions, or properties to reach logical
____________________________________________________
conclusions.
____________________________________________________
2.
Sample
answer: One valid form of deductive
____________________________________________________
reasoning
is the Law of Detachment.
____________________________________________________
Law of Syllogism
_______________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
deductive reasoning
_______________________
Law of Detachment
_______________________
New Vocabulary Write the correct term next to each
definition.
allows you to draw conclusions from two true conditional
statements when the conclusion of one statement is the
hypothesis of the other
uses facts, rules, definitions, or properties to reach logical
conclusions from given statements
form of deductive reasoning that states that if all the facts
are true, then the conclusion reached is also true
Vocabulary Link Syllogism is a word used in everyday
English as well as in mathematics. Look up syllogism in the
dictionary. Explain how the English definition can help you
remember how syllogism is used in mathematics.
Sample answer: logic; consists of major, minor
————————————————————————————
premise, and conclusion; is analogy to drawing
————————————————————————————
conclusion based on two true conditional statements
————————————————————————————
————————————————————————————
Chapter 2
27
Glencoe Geometry
Lesson 2-4
Active Vocabulary
NAME
DATE
PERIOD
Lesson 2-4 (continued)
Main Idea
Law of Detachment
pp. 115–117
Details
Identity the steps to validate the two conclusions by
completing the graphic organizer.
If the light bulb is broken,
the lamp will not work.
Jackson’s lamp will not work.
Jackson’s light bulb is
broken.
Identify the
hypothesis.
The light
bulb is
broken.
Identify the
conclusion.
The lamp
will not
work.
Analyze the conclusion.
Jackson’s light bulb is
broken. Invalid; the
lamp could be broken.
Law of Syllogism
pp. 117–118
Match the portions of each statement with the correct
term by drawing a line to connect the two. Then write
the true conditional statement and the valid
conclusion using the Law of Syllogism.
you buy bread
q
you walk to the store
r
you can make toast
If you walk to the store, you can buy bread. If you buy
————————————————————————————
bread, you can make toast.
————————————————————————————
Helping You Remember
A good way to remember something is to explain
it to someone else. Suppose that a classmate is having trouble remembering the Law of
Detachment. In your own words, explain what the Law of Detachment means?
Sample answer: a logical argument where if the first and second statements
—————————————————————————————————————————
are true, you can conclude that a third statement connecting the two true
—————————————————————————————————————————
statements is true
—————————————————————————————————————————
Chapter 2
28
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
p
NAME
DATE
2-5
PERIOD
Postulates and Paragraph Proofs
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in this lesson.
1.
Sample
answer: using basic postulates about
____________________________________________________
points,
lines, and planes
____________________________________________________
2.
Sample
answer: writing paragraph proofs
____________________________________________________
____________________________________________________
Active Vocabulary
postulate
New Vocabulary Fill in each blank with the correct term or
phrase.
a statement that is accepted as
true
without
proof
deductive argument
theorem
paragraph proof
a logical argument in which each statement you make is
supported by a statement that is accepted
true
a logical
given
a
chain
to what you are trying to prove
paragraph
given situation is
informal proof
axiom
Chapter 2
of statements that link the
statement or conjecture that has been
writing a
as
to explain why a
proven
conjecture
for a
true
paragraph proof
same as __________________
same as postulate
29
Lesson 2-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
proof
Glencoe Geometry
NAME
DATE
PERIOD
Lesson 2-5 (continued)
Main Idea
Points, Lines, and
Planes
pp. 125–126
Details
Write a postulate of points, lines, and planes. Then
model the postulate.
Through any three noncollinear points, there is
————————————————————————————
exactly one plane.
————————————————————————————
Sample answer:
Paragraph Proofs
pp. 126–127
Sequence the steps in the proof process by completing
the organizer.
Steps in the Proof Process
if possible.
theorem or conjecture to
Step 2: State the _________
proven
be ________.
deductive argument by forming
Step 3: Create a _____________________
logical chain of statements linking the given to
a _____________
what you are trying to prove.
statement with a reason.
Step 4: Justify each ____________
properties
Reasons include definitions, algebraic ____________,
postulates, and ___________.
theorems
proven
Step 5: State what has been ________.
Chapter 2
30
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
information draw a ________
picture
Step 1: List the given
___________________;
NAME
DATE
2-6
PERIOD
Algebraic Proof
What You’ll Learn
Skim the lesson. Write two things you already know about
algebraic proof.
1.
Sample
answer: properties of real numbers
____________________________________________________
2.
Sample
answer: how to solve an equation
____________________________________________________
____________________________________________________
Active Vocabulary
deductive reasoning
Review Vocabulary Write the definition next to the term.
(Lesson 2-4)
uses facts, rules, definitions, and properties to reach
————————————————————————————
logical conclusions from given statements
————————————————————————————
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
New Vocabulary Match the term with its definition by
drawing a line to connect the term.
algebraic proof
two-column proof
formal proof
same as formal proof
contains statements and reasons in two columns
uses a group of algebraic steps to solve problems and justify
each step.
Vocabulary Link Proof is a word that is used in everyday
English. Find the definition of proof using a dictionary.
Explain how the English definition can help you remember
how proof is used in mathematics.
Sample answer: evidence sufficient enough to
————————————————————————————
establish something as true; Evidence is the same as
————————————————————————————
justification with postulates.
————————————————————————————
Chapter 2
31
Glencoe Geometry
Lesson 2-6
____________________________________________________
NAME
DATE
PERIOD
Lesson 2-6 (continued)
Main Idea
Algebraic Proof
pp. 134–135
Details
Write a 2-column proof by completing each step.
8
1
Given: −
+x=6- −
x
3
3
1
Prove: x = 2 −
2
Statements
Reasons
8
1
−
+x=6-−
x
3
Given
3
8
1
x
3 −
+x =3 6-−
Multiplication Prop of
Equality
8 + 3x = 18 - x
Distributive Property
(3
)
(
)
3
8 + 3x + x = 18 - x + x
Simplify.
8 + 4x = 18
4x = 10
Substitution Prop of Equality
10
4
−
x=−
Division Prop of Equality
4
4
10
1
x=−
or 2 −
4
Chapter 2
2
Substitution Prop of
Equality
Match the property with its example by drawing a
line to connect the example.
Symmetric Property
If ST = WX, and WX = YZ,
then ST = YZ.
Reflective Property
If ∠C = ∠D, then ∠D = ∠C.
Transitive Property
∠A = ∠A
32
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Subtraction Prop of
Equality
8 + 4x - 8 = 18 - 8
Geometric Proof
p. 136
Addition Prop of Equality
NAME
DATE
2-7
PERIOD
Proving Segment Relationships
What You’ll Learn
Scan the text in Lesson 2-7. Write two facts you learned
about proving segment relationships.
1.
Sample
answer: You can use a ruler or segment
____________________________________________________
addition
to show congruent segments.
____________________________________________________
2.
Sample
answer: There are postulates about
____________________________________________________
congruence
segments.
____________________________________________________
inductive reasoning
Review Vocabulary Write the definition next to each term.
(Lessons 2-1 and 2-3)
reasoning that uses a number of specific examples to
————————————————————————————
arrive at a conclusion
————————————————————————————
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
conjecture
the conclusion that is reached within inductive
————————————————————————————
reasoning
————————————————————————————
counterexample
conditional statement
converse
an example that disproves a conjecture
————————————————————————————
a statement that can be written in if-then form
————————————————————————————
formed by exchanging the hypothesis and conclusion
————————————————————————————
of the conditional
————————————————————————————
contrapositive
formed by negating both the hypothesis and
————————————————————————————
conclusion of the converse of the conditional
————————————————————————————
inverse
formed by negating both the hypothesis and
————————————————————————————
conclusion of the conditional
————————————————————————————
Chapter 2
33
Glencoe Geometry
Lesson 2-7
Active Vocabulary
NAME
DATE
PERIOD
Lesson 2-7 (continued)
Main Idea
Ruler Postulate
pp. 142–143
Details
Use the Model and Fill in each blank to summarize the
Segment Addition Postulate.
collinear then point __
Q is between __
If P, Q, and R are __________,
P
R if and only if ____
PQ + ____
QR = PR.
and __
12
23
1
2
3
13
Segment Congruence
pp. 143–144
Complete the graphic organizer to prove the
statement below.
−−− −− −− −−−
−−− −−−
Given: QR " ST; ST " UV Prove: QR " UV
−−−
−−
−−
−−−
QR # ST and ST # UV
−− −−−
QR # UV by the
definition
of congruence .
QR = UV by the
Transitive
__________________
Property
__________________.
Helping You Remember
A good way to keep the names straight in your
mind is to associate something in the name of the postulate with the content of the
postulate. How can you use this idea to distinguish between the Rule Postulate and the
Segment Addition Postulate?
Sample answer: There are two words in Ruler Postulate, and three words in
—————————————————————————————————————————
Segment Addition Postulate. Ruler Postulate has two points in the statement,
—————————————————————————————————————————
and Segment Addition Postulate mentions three points.
—————————————————————————————————————————
Chapter 2
34
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
QR = ST and ST = UV
by the definition of
congruence
____________________.
NAME
DATE
2-8
PERIOD
Proving Angle Relationships
What You’ll Learn
Scan Lesson 2-8. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: Supplementary and
____________________________________________________
Complementary
Angles
____________________________________________________
2.
Sample
answer: Congruent Right Angles
____________________________________________________
____________________________________________________
Active Vocabulary
Review Vocabulary Fill in each blank with the correct term or
phrase. (Lessons 1-4 and 1-5)
complementary angles
adjacent angles with measures that have a sum of
two __________
90 degrees
_____________
supplemetary angles
side
angles with a common ______
adjacent angles with measures that have a sum of
two __________
180 degrees
______________
interior
inside an angle
the region of a plane _______
exterior
outside an angle
the region of a plane _________
acute angle
obtuse angle
Chapter 2
less than 90 degrees
an angle with a measure of _______________________
than 90 degrees
an angle with a measure of greater
_________________________
35
Glencoe Geometry
Lesson 2-8
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
adjacent angles
NAME
DATE
PERIOD
Lesson 2-8 (continued)
Main Idea
Supplementary and
Complementary Angles
pp. 149–150
Details
Compare and contrast the Angle Addition Postulate
and the Segment Addition Postulate by completing
the Venn diagram.
Angle Addition Postulate
add
adjacent
angles
Congruent Angles
pp. 151–153
Segment Addition Postulate
sum of
add
parts is
measure collinear
of larger segments
figure
Write a two-column proof to prove the statement by
completing the chart.
Given: m∠2 = 90
1 2 3
∠1 " ∠3
Prove: m∠1 = 45
1. m∠2 = 90
∠1 " ∠3
2. m∠3 = m∠1
1. Given
2. Definition of
congruent
3. m∠1 + m∠3 + m∠2 =
3. Definition of
complementary angles
4. m∠1 + m∠1 + 90 = 180
4. Substitution
180
5. m∠1 + m∠3 + 90 – 90
= 180 – 90
5. Subtraction Property
6. m∠1 + m∠1 = 90
6. Substitution
m∠1 + m∠1
= 90 ÷ 2
7. −
7. Division Property
8. m∠1 = 45
8. Substitution
2
Chapter 2
Reasons
36
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Statements
NAME
DATE
PERIOD
Reasoning and Proof
Tie It Together
Complete the graphic organizer with a term from the chapter.
the process of justifying
algebraic steps with
properties of real numbers
algebraic proof
the process of justifying
geometric statements with
postulates or theores
geometric proof
If p -> q is a true statement, and p
is true, then q is true.
Law of Detachment
the process of
justifying statements
based on past
observations
the process of justifying
statements with
previously accepted
statements
deductive reasoning
If p -> q and q -> r, then p -> r.
Law of Syllogism
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
inductive reasoning
Logic
p∧q
conjunction
~p
negation
If…
hypothesis
~p -> p
inverse
Chapter 2
Statements
If-Then Statements p -> q
conditional statements
q -> p
q∨p
disjunction
…Then
conclusion
~q -> ~p
converse
contrapositive
37
Glencoe Geometry
NAME
DATE
PERIOD
Reasoning and Proof
Before the Test
Now that you have read and worked through the chapter, think about what you have
learned and complete the table below. Compare your previous answers with these.
1. Write an A if you agree with the statement.
2. Write a D if you disagree with the statement.
Reasoning and Proof
After You Read
• Listing a counterexample is a method to prove a
conjecture is true.
• A Venn diagram can illustrate a conjunction.
• A form of inductive reasoning is the Law of Detachment.
• The statement a line contains two points, is an example
of a postulate.
• Supplementary angles have measures whose sum is 90°.
D
A
D
A
D
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 2 Study Guide and Review in the textbook.
□ I took the Chapter 2 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• Make up acronyms to remember lists or sequences. PEMDAS is one acronym for
remembering the order of operations (parentheses, exponents, multiply and divide, add
and subtract). (Please Excuse My Dear Aunt Sally).
Chapter 2
38
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 2.
NAME
DATE
PERIOD
Parallel and Perpendicular Lines
Before You Read
Before you read the chapter, think about what you know about the topic. List three things
you already know about parallel and perpendicular lines in the first column. Then list three
things you would like to learn about them in the second column.
K
W
What I know…
What I want to find out…
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
See students’ work.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• When you take notes, preview the lesson and make generalizations about what
you think you will learn.
Then compare that with what you actually learned after each lesson.
• Before each lesson, skim through the lesson and write any questions that
come to mind in your notes.
As you work through the lesson, record the answer to your question.
Chapter 3
39
Glencoe Geometry
NAME
DATE
PERIOD
Parallel and Perpendicular Lines
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on parallel lines and transversals, one fact might be parallel
planes are planes that do not intersect. After completing the chapter, you can use this table
to review for your chapter test.
Lesson
3-1 Parallel Lines and Transversals
Fact
See students’ work.
3-2 Angles and Parallel Lines
3-3 Adding and Subtracting Polynomials
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3-4 Equations of Lines
3-5 Proving Lines Parallel
3-6 Perpendiculars and Distance
Chapter 3
40
Glencoe Geometry
NAME
DATE
3-1
PERIOD
Parallel Lines and Transversals
What You’ll Learn
Skim the Examples for Lesson 3-1. Predict two things you
think you will learn about parallel lines and transversals.
1.
Sample
answer: how to identify and classify
____________________________________________________
2.
Sample
answer: how to classify angle pairs
____________________________________________________
____________________________________________________
Active Vocabulary
parallel lines
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
transversal
New Vocabulary Use the figure to complete each statement.
t
t is a transversal
∠ 1 and ∠ 7 are
1
.
exterior angles
3
.
interior angles
∠ 2 and ∠ 6 are corresponding angles .
exterior angles
∠ 4 and ∠ 5 are alternate interior angles
consecutive interior
angles
alternative interior angles
alternate exterior angles
corresponding angles
5
4
6
7 8
2
a
b
.
a and b are parallel lines .
∠ 4 and ∠ 6 are consecutive interior angles
.
∠ 2 and ∠ 7 are alternate exterior angles .
∠ 5 and ∠ 6 are interior angles .
Vocabulary Link Skew is a word used in everyday English as
well as in mathematics. Look up skew in the dictionary.
Explain how the English definition can help you remember
how skew is used in mathematics.
Sample answer: turn aside or swerve; Skew lines look
————————————————————————————
like they turn aside from one another.
————————————————————————————
Chapter 3
41
Glencoe Geometry
Lesson 3-1
parallels
and transversals
____________________________________________________
NAME
DATE
PERIOD
Lesson 3-1 (continued)
Main Idea
Relationships Between
Lines and Planes
pp. 171–172
Details
Summarize information about parallel lines and skew
lines in the graphic organizer below.
What are
parallel
lines?
coplanar
lines that do
not intersect
Parallel
Lines
Both types of lines
never intersect.
How are parallel
lines and skew
lines different?
Skew
Lines
What are
skew lines?
noncoplanar
lines that do
not intersect
Parallel lines are
on the same plane
and there is a
symbol, ||. Skew
lines are on
different planes
and there is no
symbol.
Model a transversal t which intersects two or more
parallel lines. Identify consecutive interior angles,
alternative interior and exterior angles, and
corresponding angles.
consecutive interior angles:
∠ 3 and ∠ 5, ∠ 4 and ∠ 6
alternate interior angles:
∠ 3 and ∠ 6, ∠ 4 and ∠ 5
alternate exterior angles:
∠ 1 and ∠ 8, ∠ 2 and ∠ 7
corresponding angles:
∠ 1 and ∠ 5, ∠ 2 and ∠ 6,
∠ 3 and ∠ 7, ∠ 4 and ∠ 8
t
1
3
5
2
4
6
7 8
a
b
Helping You Remember
Look up meaning of the prefix trans- in the
dictionary. Write down 4 words that have trans- as a prefix. How can the meaning of the
prefix help you remember the meaning of transversal?
Sample answer: Trans means across; transatlantic, Transamerica, transport,
—————————————————————————————————————————
transparent; A transversal cuts across two or more parallel lines.
—————————————————————————————————————————
Chapter 3
42
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Transversal Angle Pair
Relationships
pp. 172–173
How are parallel
lines and skew
lines similar?
NAME
DATE
3-2
PERIOD
Angles and Parallel Lines
What You’ll Learn
Scan Lesson 3-2. List two headings you would use to make
an outline of this heading.
1.
Sample
answer: Parallel Lines and Angle Pairs
____________________________________________________
____________________________________________________
2.
Sample
answer: Algebra and Angle Measures
____________________________________________________
Active Vocabulary
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
parallel lines
skew
Review Vocabulary Match the term with the correct
definition by drawing a line between the two. (Lesson 3-1)
lie in the regions not between parallel lines that are cut by a
transversal
a line that intersects 2 or more coplanar lines at 2 different
points
interior angles
lie on the same side of both the parallel lines and the
transversal
exterior angles
coplanar lines that never intersect
consecutive interior angles
lie between parallel lines that are cut by a transversal
corresponding angles
interior lines that lie on the same side of a transversal
tranversal
Chapter 3
lines that are not on the same plane and never intersect
43
Glencoe Geometry
Lesson 3-2
____________________________________________________
NAME
DATE
PERIOD
Lesson 3-2 (continued)
Main Idea
Details
Parallel Lines and Angle
Pairs
pp. 178–179
Fill in the organizer about the Corresponding Angles
Postulate.
Corresponding
Angles Postulate
Why is it useful?
Definition
If two parallel lines are
cut by a transversal, each
pair of corresponding
angles is congruent.
If you know an angle
measure, you will know
that the corresponding
angle has the same
measure. Consecutive
interior angles will be
supplementary.
Example
Nonexample
Sample answer:
t
1
3
5
2
4
6
7 8
Sample answer:
t
q
5
p
4
3
6
7 8
q
∠ 3 and ∠ 7 are not
congruent because p and
q are not parallel.
Fill in each blank to summarize the Perpendicular
Transversal Theorem.
The Perpendicular Transversal Theorem states that in a
plane , if a line is perpendicular to one of two
parallel lines , then it is perpendicular to the other.
Helping You Remember
How can you use an everyday meaning of the
adjective alternate to help you remember the types of angle pairs for two lines and a
transversal?
Sample answer: One definition of alternate says first on one side and then the
—————————————————————————————————————————
other. Alternating angles occur on opposite sides of a transversal.
—————————————————————————————————————————
Chapter 3
44
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
∠ 3 " ∠ 7 because they
are corresponding
angles.
Algebra and Angle
Measures
pp. 180
1 2
p
NAME
3-3
DATE
PERIOD
Slopes of Lines
What You’ll Learn
Scan the text in Lesson 3-3. Write two facts you learned
about slopes of lines as you scanned the text.
1.
Sample
answer: Slopes can be positive, negative,
____________________________________________________
zero,
or undefined.
____________________________________________________
2.
Sample
answer: Parallel lines have equal slopes.
____________________________________________________
____________________________________________________
line
point
defined term
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
undefined term
Review Vocabulary Fill in each blank with the correct term or
phrase. (Lesson 1-1)
made up of
a location
points with no thickness or width
without shape or size
can be explained using undefined
terms
can only be explained using examples and descriptions
New Vocabulary Write the definition next to each term.
slope
ratio of the change along the y-axis to the change
————————————————————————————
along the x-axis between any two points on the line
————————————————————————————
————————————————————————————
————————————————————————————
rate of change
same as slope; describes how a quantity y changes in
————————————————————————————
relationship to a quantity x
————————————————————————————
————————————————————————————
————————————————————————————
Chapter 3
45
Glencoe Geometry
Lesson 3-3
Active Vocabulary
NAME
DATE
PERIOD
Lesson 3-3 (continued)
Main Idea
Slope of a Line
pp. 186–188
Details
Complete the organizer to summarize the steps of
finding the slope of a line. Then fill in each blank of
the example.
Find the Slope of a Line
Step 1:
Label the cordinates as
(x1, y1) and (x2, y2).
Step 2:
Substitute the
coordinates into the
slope formula.
y2 - y1
m=−
x2 - x1
Parallel and
Perpendicular Lines
pp. 189–190
5 y
4
3
2
1
0
x
1 2 3 4 5
-5-4-3 -2-1
-1
-2
-3
-4
-5
Step 1: Substitute (-2, -1) for
(x1, y1) and (2, 5) for (x2, y2).
Step 2: Use the formula.
Step 3: Simplify.
5 - (-1)
2 - (-2)
−
3
6
=−
−
4
2
Fill in each blank for the Parallel and Perpendicular
Lines Postulates.
Slopes of Parallel Lines: Two non-vertical lines
have the same
will
slope if they are parallel . Vertical lines
are always parallel.
nonvertical
lines are perpendicular if the product of their
slopes is -1 . Vertical and horizontal lines are always
Slopes of Perpendicular Lines: Two
perpendicular.
Helping You Remember
Suppose 2 non-vertical lines have slopes which
are reciprocals of each other. Are they perpendicular? Explain.
Sample answer: Reciprocals have a product of 1, not -1 so the lines are not
—————————————————————————————————————————
perpendicular.
—————————————————————————————————————————
Chapter 3
46
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Step 3: Simplify.
Find the slope of a line.
NAME
DATE
3-4
PERIOD
Equations of Lines
What You’ll Learn
Skim Lesson 3-4. Predict two things that you expect to learn
based on the headings and the Key Concept Box.
1.
Sample
answer: how to write equations of lines
____________________________________________________
____________________________________________________
2.
Sample
answer: how to solve real-world problems
____________________________________________________
by
writing equations
____________________________________________________
Active Vocabulary
postulate
proof
theorem
a logical chain of statements that link the given to what you
are trying to prove
a statement or conjecture that has been proven
a logical argument in which each statement you make is
supported by a statement that is accepted as true
a statement that is accepted as true without proof
New Vocabulary Write the correct equation next to each term.
point-slope form
slope-intercept form
y = mx + b
y - y1 = m(x - x1)
Vocabulary Link Slope is a word used in everyday English as
well as in mathematics. Look up slope in the dictionary.
Explain how the English definition can help you remember
how slope is used in mathematics.
Sample answer: steepness; The slope of a line is the
————————————————————————————
steepness of the line.
————————————————————————————
————————————————————————————
————————————————————————————
Chapter 3
47
Glencoe Geometry
Lesson 3-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
deductive argument
Review Vocabulary Match the term with its definition by
drawing a line to connect the two. (Lesson 2-5)
NAME
DATE
PERIOD
Lesson 3-4 (continued)
Main Idea
Write Equations of Lines
pp. 196–198
Details
Complete the chart on how to write an equation of a
line.
Writing an Equation of a Line
Forms
Procedures
Given:
slope and y-intercept
Given:
slope and point on the
line
Given: two points
Write Equations to Solve
Problems
p. 199
Sample answer: Solve
the equation for y, then
find m. Use the opposite
reciprocal of m to write
the equation.
Fill in each blank to write the equations.
Alicia wants to paint for about 5 hours. One art studio
charges an $8 fee plus $2 per hour. The second studio does
not charge a fee and charges $4 an hour. Write two equations
to represent the cost of each studio. Which studio should she
choose?
Fee + Hourly fee × x for Number of Hours = Total
Cost
8 +
2 x=y
0
4
+
x=y
The first studio charges a better deal for up to 4 hours.
Chapter 3
48
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Given:
for a horizontal line,
when m = 0
Given: equation of one
line and one set of
points for ⊥ and " lines
Sample answer: Use
slope-intercept form.
Substitute m for slope
and b for the y-intercept.
Sample answer: Use
point-slope form.
Substitute m for slope
and a point for (x1, y1)
and then solve for y.
Sample answer: Use the
slope formula to find m,
then use point-slope
form.
Sample answer: Use the
point-slope form where
y = b.
NAME
DATE
3-5
PERIOD
Proving Lines Parallel
What You’ll Learn
Skim the lesson. Write two things you already know about
proving lines parallel.
1.
Sample
answer: Parallel lines have the same
____________________________________________________
slope.
____________________________________________________
____________________________________________________
2.
Sample
answer: Corresponding angles are
____________________________________________________
congruent,
so knowing their measures could
____________________________________________________
prove
lines parallel.
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
alternative interior
angles
alternate exterior
angles
corresponding angles
parallel lines
transversal
interior angles
exterior angles
Chapter 3
Review Vocabulary Write the correct term next to each
definition. (Lesson 3-1)
nonadjacent interior angles that lie on opposite sides of a
transversal
nonadjacent exterior angles that lie on opposite sides of a
transversal
angles that lie on the same side of the transversal and on
the same side of parallel lines
coplanar lines that do not intersect
line that intersects two or more parallel lines at two
different points
Where a transversal cuts through two parallel lines, these
angles lie in the same region between parallel lines.
Where a transversal cuts through two parallel lines, these
angles lie in the same two regions that are not between the
two parallel lines.
49
Glencoe Geometry
Lesson 3-5
Active Vocabulary
NAME
DATE
PERIOD
Lesson 3-5 (continued)
Main Idea
Identify Parallel Lines
pp. 205–207
Details
Use the model to cross out the congruence statement
in the concept circle that does not belong.
1
5
Prove Lines Parallel
p. 208
3
2
6
7
∠1 " ∠8
∠2 " ∠4
∠6 " ∠7
∠2 " ∠7
4
8
Fill in each blank to summarize the lesson about
proving lines are parallel.
When two
parallel
lines are cut by a transversal
congruent
or
supplementary . When a pair of lines form angles that
do not meet this condition, the lines cannot be parallel.
Helping You Remember
A good way to remember something new is to
draw a picture. How can a sketch help you to remember the Parallel Postulate?
Sample answer: Draw a line with a ruler and choose a point not on the line. Try
—————————————————————————————————————————
to draw lines through the point that are parallel, and you will see there is only
—————————————————————————————————————————
one way to do this.
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 3
50
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
the angle pairs formed are either
,
NAME
3-6
DATE
PERIOD
Perpendiculars and Distance
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in this lesson.
1.
Sample
answer: how to find the distance between
____________________________________________________
2.
Sample
answer: how to find the distance between
____________________________________________________
parallel
lines
____________________________________________________
Active Vocabulary
parallel lines
skew
Review Vocabulary Write the definition next to each term.
(Lesson 2-4)
collinear lines that never intersect
————————————————————————————
noncoplanar lines that never intersect
————————————————————————————
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
New Vocabulary Fill in each blank with the correct term or
phrase.
equidistance
The distance
between two
measured along a
parallel
perpendicular
lines when
line to the lines is
always the same .
Vocabulary Link Equidistance is a word that is used in
everyday English. Find the definition of equidistance using a
dictionary. Explain how the English definition can help you
remember how equidistance is used in mathematics.
Sample answer: always the same distance;
————————————————————————————
Equidistance means the same thing in mathematics
————————————————————————————
and everyday life.
————————————————————————————
————————————————————————————
Chapter 3
51
Glencoe Geometry
Lesson 3-6
a____________________________________________________
line and point
NAME
DATE
PERIOD
Lesson 3-6 (continued)
Main Idea
Distance from a Point to
a Line
pp. 213–216
Details
Summarize the steps to find the distance between a
point p and a line ℓ on a coordinate plane.
Find the Distance between a Point and a Line.
Step 1: Write an equation for the line ℓ.
Step 2: Write an equation for the perpendicular line
that goes through point p
Step 3: Use system of equation to find the
coordinates where line ℓ intersects
perpendicular line.
the
Step 4: Use the Distance Formula to find the
distance between point P and the point of intersection of
the two lines.
Fill in each blank to find the distance between the
parallel lines.
The lines are 11 units apart.
1. y = 3x – 7
y = 3x + 4
2. y = –2x – 8
The lines are 10 units apart.
y = –2x + 2
Helping You Remember
Use your dictionary to find the meaning of the
Latin root aequus. List three words that are derived from this root and give meaning to
each.
Sample answer: Aequus means even, fair, or equal. Equinox means one of the
————————————————————————————————————————
two times a year when day and night are of equal length. Equity means being
————————————————————————————————————————
just or fair. Equivalent means being equal in value or meaning.
————————————————————————————————————————
Chapter 3
52
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Distance Between
Parallel Lines
pp. 216–217
NAME
DATE
PERIOD
Parallel and Perpendicular Lines
Tie It Together
Complete the graphic organizer with a term or formula from the chapter.
vertical line
horizontal line
x=a
y=a
point-slope form
opposite
y - y1 = m(x - x1)
reciprocal
slopes
slope-intercept form
perpendicular
equations
y = mx + b
equal
slopes
parallel
positive
y
x
0
skew lines
lines that never
intersect and
are not
coplanar
lines
slope
negative
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
consecutive
interior angles
parallel
perpendicular
coplanar
lines that
never
intersect
coplanar lines
that intersect
to form a right
angle
x
0
zero
y
0
alternate
interior angles
alternate
exterior angles
corresponding
angles
undefined
y
0
Chapter 3
53
x
Glencoe Geometry
x
NAME
DATE
PERIOD
Parallel and Perpendicular Lines
Before the Test
Review the ideas you listed in the table at the beginning of the chapter. Cross out any
incorrect information in the first column. Then complete the table by filling in the third
column.
K
W
L
What I know…
What I want to find out…
What I learned…
See students’ work.
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 3 Study Guide and Review in the textbook.
□ I took the Chapter 3 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• Designate a place to study at home that is free of clutter and distraction. Try to study at
about the same time each afternoon or evening so that it is part of your routine.
Chapter 3
54
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 3.
NAME
DATE
PERIOD
Congruent Triangles
Before You Read
Before you read the chapter, think about what you know about congruent triangles. List
three things you already know about them in the first column. Then list three things you
would like to learn about them in the second column.
K
W
What I know…
What I want to find out…
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
See students’ work.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• Remember to always take notes on your own.
Don’t use someone else’s notes as they may not make sense.
• When you take notes, listen or read for main ideas.
Then record those ideas in a simplified form for future reference.
Chapter 4
55
Glencoe Geometry
NAME
DATE
PERIOD
Congruent Triangles
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on congruent triangle, one fact might be that in two congruent
triangles all of the side and angles of one triangle are congruent to the corresponding parts
of the other triangle. After completing the chapter, you can use this table to review for your
chapter test.
Lesson
4-1 Classifying Triangles
Fact
See students’ work.
4-2 Angles of Triangles
4-3 Congruent Triangles
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4-4 Proving Triangles Congruent–SSS,
SAS
4-5 Proving Triangles Congruent–ASA,
AAS
4-6 Isosceles and Equilateral Triangles
4-7 Congruence Transformations
4-8 Triangles and Coordinate Proof
Chapter 4
56
Glencoe Geometry
NAME
DATE
4-1
PERIOD
Classifying Triangles
What You’ll Learn
Scan Lesson 4-1. Predict two things that you expect to learn
based on the headings and the Key Concept box.
1.
Sample
answer: how to classify triangles by their
____________________________________________________
2.
Sample
answer: how to classify triangles by their
____________________________________________________
angles
____________________________________________________
Active Vocabulary
New Vocabulary Label the diagram with the correct terms.
acute triangle
72°
equiangular triangle
58°
equilateral triangle
50°
isosceles triangle
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
obtuse triangle
isosceles triangle
acute triangle
right triangle
right triangle
60°
scalene triangle
120°
60°
60°
equiangular triangle
11
obtuse triangle
7
13
scalene triangle
Chapter 4
57
equilateral triangle
Glencoe Geometry
Lesson 4-1
sides
____________________________________________________
NAME
DATE
PERIOD
Lesson 4-1 (continued)
Main Idea
Classify Triangles by
Angles
pp. 235–236
Details
Classify the triangle below as acute, equiangular,
obtuse, or right.
Acute: Does the triangle have 3
90°
no
acute angles?
Equiangular: Does the triangle
have 3 congruent acute angles?
no
Obtuse: Does the triangle have an
angle greater than 90˚?
right triangle
no
Right: Does the triangle have 1
right angle?
Classify Triangles by
Sides
pp. 236–237
yes
Classify each triangle below as equilateral, isosceles,
or scalene.
2 in.
2 in.
2 in.
12 cm
isosceles
equilateral
Helping You Remember
A good way to remember a new mathematical
term is to relate it to a nonmathematical definition of the same word. How is the use of
the word acute, when used to describe acute pain, related to the use of the word acute
when used to describe an acute angle or an acute triangle?
Sample answer: An acute pain is a sharp, focused pain. An acute angle is less
—————————————————————————————————————————
than 90˚, so you can think of it as a sharp, focused angle that is not spread out.
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 4
58
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7 cm
12 cm
NAME
4-2
DATE
PERIOD
Angles of Triangles
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in the lesson.
1.
Sample
answer: how to apply the Triangle
____________________________________________________
Angle-Sum
Theorem
____________________________________________________
2.
Sample
answer: how to apply the Exterior Angle
____________________________________________________
Theorem
____________________________________________________
auxiliary line
remote interior angle
exterior angle
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
flow proof
New Vocabulary Write the correct term next to each
definition.
an extra line or segment drawn in a figure to help analyze
geometric relationships
one of two angles inside a triangle that is not adjacent to an
angle outside the triangle
an angle formed by one side of a triangle and the extension
of an adjacent side
a method of showing statements to be true using boxes and
arrows to show the logical progression of an argument
Vocabulary Link Auxiliary is a word that is used in everyday
English. Find the definition of auxiliary using a dictionary.
Explain how the English definition can help you remember
how auxiliary line is used in mathematics.
Sample answer: Auxiliary means offering or providing
————————————————————————————
help; giving assistance or support. So, it makes sense
————————————————————————————
that an auxiliary line is a line drawn in a figure to
————————————————————————————
help you analyze geometric relationships.
————————————————————————————
————————————————————————————
Chapter 4
59
Glencoe Geometry
Lesson 4-2
Active Vocabulary
NAME
DATE
PERIOD
Lesson 4-2 (continued)
Main Idea
Triangle Angle-Sum
Theorem
pp. 244–245
Details
Fill in each box to complete the flow proof.
"
Given: ∠A " ∠D
∠B " ∠E
Prove: ∠C " ∠F
#
Proof:
∠A " ∠D
&
$
'
∠B " ∠E Given
Given
m∠A + m∠B + m∠C = 180
%
m∠D + m∠E + m∠F = 180
Angle-Sum
Theorem
Angle-Sum
Theorem
m∠A + m∠B + m∠C = m∠D + m∠E + m∠F
Substitution
Substitution
m∠C = m∠F
∠C = ∠F
Exterior Angle Theorem
pp. 246–247
Subtraction
Def. of congruent angles
Solve for angles 1 and 2 in the figure below.
70°
m∠1 = 115
45°
2
1
m∠2 = 65
Chapter 4
60
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
m∠A + m∠B + m∠C = m∠A + m∠B + m∠F
NAME
DATE
4-3
PERIOD
Congruent Triangles
What You’ll Learn
Scan the lesson. Write two things you already know about
congruent triangles.
1.
Sample
answer: The corresponding parts of
____________________________________________________
congruent
triangles are congruent.
____________________________________________________
____________________________________________________
2.
Sample
answer: If the corresponding parts of two
____________________________________________________
triangles
are congruent, then the triangles are
____________________________________________________
congruent.
____________________________________________________
Active Vocabulary
Review Vocabulary Write the definition next to each term.
(Lesson 4-1)
a triangle with 3 congruent acute angles
isosceles triangle
a triangle with at least 2 congruent sides
scalene triangle
————————————————————————————
————————————————————————————
a triangle with no congruent sides
————————————————————————————
New Vocabulary Fill in each blank with the correct term or
phrase.
If two geometric figures have exactly the same shape and
congruent
size, they are ______________.
In two
congruent polygons
polygon are congruent to the
, all of the parts of one
corresponding parts
or
matching parts of the other polygon.
Chapter 4
61
Glencoe Geometry
Lesson 4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
equiangular triangle
NAME
DATE
PERIOD
Lesson 4-3 (continued)
Main Idea
Congruent and
Corresponding Parts
pp. 253–254
Details
If !ABC " !DEF, complete the diagram to solve for x.
"
'
21
#
$
Write a congruence
relationship that will
help solve the problem.
# #
DF ! AC
Prove Triangles
Congruent
pp. 255–256
9x - 4
32
40
&
%
Use the congruence
relationship to
to write an equation.
Solve the equation for x.
9x - 4 = 32
x=4
Check
Name each property of triangle congruence.
If !ABC " !DEF, then !DEF " !ABC.
!RST " !RST
Reflexive Property
If !ABC " !DEF and !DEF " !LMN, then
!ABC " !LMN.
Transitive Property
Helping You Remember
Suppose a classmate is having trouble writing
congruence statements for triangles because he thinks he has to match up three pairs of
sides and three pairs of angles. How can you help him understand how to write correct
congruence statements more easily?
Sample answer: Name one of the triangles using its three vertices. Name
—————————————————————————————————————————
the other by listing corresponding vertices in the same order as the first.
—————————————————————————————————————————
Chapter 4
62
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Symmetric Property
NAME
4-4
DATE
PERIOD
Proving Triangles Congruent – SSS, SAS
What You’ll Learn
Scan Lesson 4-4. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: SSS
____________________________________________________
____________________________________________________
2.
Sample
answer: SAS
____________________________________________________
____________________________________________________
Active Vocabulary
Review Vocabulary Use the diagram to write an example for
each of the following. (Lesson 4-2)
:
8
;
auxiliary line
two remote interior angles
to angle XYZ
exterior angle
"
Sample answer: 
YZ
————————————————————————————
Sample answer: ∠W and ∠X
————————————————————————————
Sample answer: ∠XYZ
————————————————————————————
New Vocabulary Write the definition next to the term.
included angle
the angle formed by two adjacent sides of a polygon
————————————————————————————
————————————————————————————
————————————————————————————
————————————————————————————
Chapter 4
63
Glencoe Geometry
Lesson 4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9
NAME
DATE
PERIOD
Lesson 4-4 (continued)
Main Idea
SSS Postulate
pp. 262–264
Details
Use the coordinate grid below to graph two triangles
that are congruent. Explain how you could show they
are congruent using the SSS postulate. Then show
they are equivalent.
8
6
4
2
-8
-6
-4
2
-2
6
4
8
-2
-4
-6
-8
!ABC
!DEF
,
)
D(
,
)
B(
,
)
E(
,
)
C(
,
)
F(
,
)
Sample answer: Use the vertices of the triangles and
————————————————————————————
the distance formula to find the length of each side.
————————————————————————————
If the corresponding sides have the same lengths, the
————————————————————————————
triangles are congruent by the SSS postulate.
————————————————————————————
SAS Postulate
pp. 264–266
Use the SAS Postulate to write a triangle congruence
statement for the figure below.
2
6
!QRS " !TUS
_____________________
3
Chapter 4
64
4
5
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A(
NAME
4-5
DATE
PERIOD
Proving Triangles Congruent – ASA, AAS
What You’ll Learn
Skim the Examples for Lesson 4-5. Predict two things you
think you will learn about proving triangles congruent.
1.
Sample
answer: how to prove triangles congruent
____________________________________________________
using
the ASA Postulate
____________________________________________________
____________________________________________________
2.
Sample
answer: how to prove triangles congruent
____________________________________________________
using
the AAS Theorem
____________________________________________________
____________________________________________________
Active Vocabulary
Review Vocabulary Write the name of the postulate that
could be used to prove !EFG " !EHG. (Lesson 4-4)
SAS Postulate
________________________
'
)
(
&
10 cm
SSS Postulate
________________________
'
6 cm
10 cm
(
6 cm
)
New Vocabulary Write the correct term next to the definition.
included side
________________________
Chapter 4
the side located between two consecutive angles of a polygon
65
Glencoe Geometry
Lesson 4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
&
NAME
DATE
PERIOD
Lesson 4-5 (continued)
Main Idea
ASA Postulate
pp. 273–274
Details
Fill in each blank with the appropriate terms in the
following statements.
6
/
70°
#
50°
50°
"
70°
$
,
In order to apply the ASA Postulate to prove two triangles
are congruent, two pairs of corresponding angles
included side
and the
of one triangle must be
congruent
to the corresponding parts of
a second triangle.
AAS Theorem
pp. 274–276
Use the AAS Theorem to write a triangle congruence
statement for the figure below.
1
71°
75°
!PRS " !SYP
_________________
3
:
75°
71°
4
Helping You Remember
Summarize what is needed to prove triangle
congruence using the four methods learned so far.
SSS
Three pairs of
corresponding
sides are
congruent.
Chapter 4
SAS
Two pairs of
corresponding
sides and their
included angles
are congruent.
ASA
Two pairs of
corresponding
angles and their
included sides
are congruent.
66
AAS
Two pairs of
corresponding
angles and their
corresponding
non-included
sides are
congruent.
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
In the figure above, because ∠ A " ∠ B ,
∠ N " ∠ C , and AN " BC , you know
that # UNA " # KCB .
NAME
DATE
4-6
PERIOD
Isosceles and Equilateral Triangles
What You’ll Learn
Scan the text in Lesson 4-6. Write two facts you learned
about isosceles and equilateral triangles as you scanned the
text.
1.
Sample
answer: If two sides of a triangle are
____________________________________________________
are
congruent.
____________________________________________________
2.
Sample
answer: An equilateral triangle is also an
____________________________________________________
equiangular
triangle.
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
isosceles triangle
equilateral triangle
obtuse triangle
Review Vocabulary Write the correct term next to each
definition. (Lesson 4-1)
a triangle with at least two congruent sides
a triangle with three congruent sides
a triangle with one angle greater than 90˚
New Vocabulary Label the diagram with the correct terms.
3
base angle
vertex angle
vertex angle
base angle
5
Chapter 4
67
4
Glencoe Geometry
Lesson 4-6
congruent,
then the angles opposite those sides
____________________________________________________
NAME
DATE
PERIOD
Lesson 4-6 (continued)
Main Idea
Properties of Isosceles
Triangles
pp. 283–284
Details
Describe the Isosceles Triangle Theorem and its
converse in the boxes below. Then write a conditional
statement using the figures to illustrate each
theorem.
Isosceles Triangle Theorem
If two sides of a triangle are
congruent, then the angles
opposite those sides are
congruent.
−−
-
2
1
.
/
−−
Example: If LM " LN, then ∠1 " ∠2.
Converse of Isosceles
Triangle Theorem
4
1
If two angles of a triangle are
congruent, then the sides
opposite those angles are
congruent.
2
5
−−
Example: If ∠1 " ∠2, then RS " RT.
Properties of Equilateral
Triangles
pp. 284–286
Solve for x in equilateral triangle QRS.
2
1
x = 10 −
3
__________________
3
(6x - 2)°
4
Helping You Remember
If a theorem and its converse are both true, you
can often remember them most easily by combining them into an “if-and-only-if”
statement. Write such a statement for the Isosceles Triangle Theorem and its converse.
Sample answer: A triangle is isosceles if and only if two angles of the triangle
—————————————————————————————————————————
are congruent.
—————————————————————————————————————————
Chapter 4
68
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
−−
3
NAME
4-7
DATE
PERIOD
Classifying Triangles
What You’ll Learn
Skim Lesson 4-7. Predict two things that you expect to learn
based on the headings and the Key Concept box.
1.
Sample
answer: how to identify real-world
____________________________________________________
transformations
____________________________________________________
____________________________________________________
2.
Sample
answer: how to verify congruence of two
____________________________________________________
fi
gures after a transformation
____________________________________________________
Active Vocabulary
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
congruence transformation
image
isometry
a transformation in which the position of the image may
differ from that of the preimage, but the two figures remain
congruent
a type of transformation that is a flip over a line
also known as a rigid transformation
preimage
the end result of a geometric transformation
reflection
an operation that maps an original geometric figure onto a
new figure
rotation
a type of transformation that is a turn around a fixed point
transformation
translation
Chapter 4
New Vocabulary Match the term with its definition by
drawing a line to connect the two.
a type of transformation that is a slide of a figure
the original figure in a geometric transformation
69
Glencoe Geometry
Lesson 4-7
____________________________________________________
NAME
DATE
PERIOD
Lesson 4-7 (continued)
Main Idea
Identify Congruence
Transformation
pp. 294–295
Details
Write the type of transformation of triangle ABC (the
preimage) in each box.
y
$!
$
translation
#!
"!
#!
"!
0
"
#
"!
#!
x
reflection
rotation
$!
$!
Verify Congruence
p. 296
Complete each statement to verify the congruence.
5
4
-4
3
4
ST =
5
2
RT =
4
y
3
-2 0
4
-2
3 x
R'S' = 3
S'T' = 5
-4
5
-6
R'T' = 4
Helping You Remember
Geometric reflections, translations, and rotations
are also known as flips, slides, and turns. Describe how these terms appropriately
illustrate the corresponding transformations.
Sample answer: In a reflection, points are flipped across a line. In a translation,
—————————————————————————————————————————
points are slid the same distance and direction. In a rotation, points are turned
—————————————————————————————————————————
about a common point.
—————————————————————————————————————————
Chapter 4
70
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-6
RS =
6
NAME
4-8
DATE
PERIOD
Triangles and Coordinate Proof
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in the lesson.
1.
Sample
answer: how to position and label
____________________________________________________
triangles
for use in coordinate proofs
____________________________________________________
____________________________________________________
2.
Sample
answer: how to write coordinate proofs
____________________________________________________
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
transformation
________________________
Review Vocabulary Write the correct term next to each
definition. (Lesson 4-7)
an operation that maps an original geometric figure onto a
new figure
translation
________________________
a type of transformation that is a slide of a figure
isometry
________________________
also known as a rigid transformation
reflection
________________________
a type of transformation that is a flip over a line
image
________________________
the end result of a geometric transformation
rotation
________________________
a type of transformation that is a turn around a fixed point
preimage
________________________
the original figure in a geometric transformation
New Vocabulary Write the definition next to the term.
coordinate proof
a method of using figures in the coordinate plane
————————————————————————————
and algebra to prove geometric concepts
————————————————————————————
————————————————————————————
————————————————————————————
Chapter 4
71
Glencoe Geometry
Lesson 4-8
Active Vocabulary
NAME
DATE
PERIOD
Lesson 4-8 (continued)
Main Idea
Position and Label
Triangles
pp. 301–302
Details
Complete the boxes below to illustrate the four key
steps in placing a triangle on a coordinate grid to aid
in coordinate proof.
Step 4Use
coordinates
y
that make
6
computations
simple.
4
Step 1Use
the origin as a
vertex.
Step 3
Keep the
triangle in
the first
quadrant
$
2
x
"
2
4
6#
Step 2Place at least
one side on an axis.
Write Coordinate Proofs
pp. 302–303
Complete each statement showing the side lengths of
triangle RST. Round to the nearest hundredth if
necessary. Classify the triangle.
RS ! 6.71
y
5
RT ! 6.32
TS !
4
x
3 1 2 3 4 5 6 7
5
Triangle RST is scalene .
Helping You Remember
Many students find it easier to remember
mathematical formulas if they can put them into words in a compact way. Describe how
you can use coordinate proof to help you remember the midpoint formula.
Sample answer: When you plot two points on the coordinate grid, the midpoint
—————————————————————————————————————————
is halfway between the x- and y-coordinates of the endpoints. Add the
—————————————————————————————————————————
x-coordinates and add the y-coordinates. Divide each by 2.
—————————————————————————————————————————
Chapter 4
72
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7
6
5
4
3
2
1
NAME
DATE
PERIOD
Congruent Triangles
Tie It Together
Complete the graphic organizer with a term or formula from the chapter.
Obtuse
Acute
Right
Equilateral
Isosceles
Equiangular
Scalene
Translations
Classified by
Angles
Classified
by Sides
Interior
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Rotations
Transformations
Types
Sum of 180°
Angles
Triangles
Exterior
Reflections
Congruence
Exterior = Sum
of remote
interiors
Sum of 360°
Reflexive
Properties
SSS
Theorems
Symmetric
SAS
Transitive
AAS
ASA
Chapter 4
73
Glencoe Geometry
NAME
DATE
PERIOD
Congruent Triangles
Before the Test
Review the ideas you listed in the table at the beginning of the chapter. Cross out any
incorrect information in the first column. Then complete the table by filling in the third
column.
K
W
L
What I know…
What I want to find out…
What I learned…
See students’ work.
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 4 Study Guide and Review in the textbook.
□ I took the Chapter 4 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• Be an active listener in class. Take notes, circle or highlight information that your
teacher stresses, and ask questions when ideas are unclear to you.
Chapter 4
74
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 4.
NAME
DATE
PERIOD
Relationships in Triangles
Before You Read
Before you read the chapter, respond to these statements.
1. Write an A if you agree with the statement.
2. Write a D if you disagree with the statement.
Before You Read
Relationships in Triangles
See students’ work.
• Concurrent lines do not intersect and
stay the same distance apart.
• To find the incenter of a triangle, draw
a circle within the triangle.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
• Every triangle has 3 medians that are
concurrent.
• The largest angle in a triangle is
opposite the longest side.
• The Hinge Theorem is a way of proving
triangle relationships.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• When you take notes, include personal experiences that relate to the lesson
and ways in which what you have learned will be used in your daily life.
• When you take notes, write questions you have about the lessons in the
margin of your notes.
Then include the answers to these questions as you work through the lesson.
Chapter 5
75
Glencoe Geometry
NAME
DATE
PERIOD
Relationships in Triangles
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on medians and altitudes of triangles, one fact might be lines
containing the altitudes of a triangle are concurrent, intersecting at a point called the
orthocenter. After completing the chapter, you can use this table to review for your chapter
test.
Lesson
5-1 Bisectors of Triangles
Fact
See students’ work.
5-2 Medians and Altitudes of Triangles
5-3 Inequalities of One Triangle
5-4 Indirect Proof
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5-5 The Triangle Inequality
5-6 Inequalities in Two Triangles
5-7 Congruence Transformations
5-8 Triangles and Coordinate Proof
Chapter 5
76
Glencoe Geometry
NAME
DATE
5-1
PERIOD
Bisectors of Triangles
What You’ll Learn
Skim Lesson 5-1. Predict two things that you expect to learn
based on the headings and figures in the lesson.
1.
Sample
answer: how to use the Perpendicular
____________________________________________________
____________________________________________________
2.
Sample
answer: how to use the Circumcenter
____________________________________________________
Theorem
____________________________________________________
____________________________________________________
Active Vocabulary
point of concurrency
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
perpendicular bisector
circumcenter
concurrent lines
incenter
New Vocabulary Write the correct term next to each
definition.
the place where three or more intersecting lines meet
a segment, line, or plane that intersects a segment at its
midpoint and is perpendicular to the segment
the point of intersection of the perpendicular bisectors of the
three sides of a triangle
three or more lines that intersect at a common point
the point of intersection of the angle bisectors of the three
angles of a triangle
Vocabulary Link Look up the definition of bisect. Use it to
write the meaning of an angle bisector.
Sample
answer: Bisect means to cut or divide into
___________________________________________________
two
equal parts. So, an angle bisector divides an
___________________________________________________
angle
into two equal angles.
___________________________________________________
Chapter 5
77
Glencoe Geometry
Lesson 5-1
Bisector
Theorems
____________________________________________________
NAME
DATE
PERIOD
Lesson 5-1 (continued)
Main Idea
Perpendicular Bisectors
pp. 322–324
Details
Describe the Perpendicular Bisector Theorem in the
box below. Then write a conditional statement using
the figures to illustrate the theorem.
Perpendicular Bisector Theorem
If a point is on the
perpendicular bisector of
a segment, then it is
equidistant from the
endpoints of the
segment.
-
1
.
2
0
−−
−−− −−
then MO = ON.
/
−−
Example: If LO is a perpendicular bisector of MN,
Angle Bisectors
pp. 324–326
Solve for x in the figure below.
x=
5x + 2
3
90°
8x - 7
Helping You Remember
A good way to remember theorems and
postulates in geometry is to explain them to other classmates in your own words. How
would you describe the Angle Bisector Theorem to a classmate who is having difficulty
understanding the theorem
Sample answer: Any point on the bisector of an angle is the same distance
—————————————————————————————————————————
from both sides of the angle.
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 5
78
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
90°
NAME
5-2
DATE
PERIOD
Medians and Altitudes of Triangles
What You’ll Learn
Skim the Examples for Lesson 5-2. Predict two things you
think you will learn about medians and altitudes of
triangles.
1.
Sample
answer: how to use the Centroid Theorem
____________________________________________________
____________________________________________________
2.
Sample
answer: how to find the orthocenter on
____________________________________________________
Active Vocabulary
Review Vocabulary Label the diagram with the correct term.
(Lesson 5-1)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
QPJOUPGDPODVSSFODZ
New Vocabulary Write the definition next to each term.
median
a
segment with endpoints being a vertex of a triangle
___________________________________________________
and
the midpoint of the opposite side
___________________________________________________
centroid
the
point of concurrency of the medians of a triangle
___________________________________________________
___________________________________________________
altitude
a
segment of a triangle from a vertex that is
___________________________________________________
perpendicular
to the line containing the opposite
___________________________________________________
of
the triangle
___________________________________________________
orthocenter
the
point of concurrency of the lines containing the
___________________________________________________
altitudes
of a triangle
___________________________________________________
Chapter 5
79
Glencoe Geometry
Lesson 5-2
a____________________________________________________
coordinate plane
NAME
DATE
PERIOD
Lesson 5-2 (continued)
Main Idea
Medians
pp. 333–335
Details
Point O is the centroid of triangle VTR. Use the
Centroid Theorem to complete each statement.
7
6
5
TO =
VS =
Altitudes
pp. 335–337
2
−
3
3
−
2
8
0
4
3
TW
WR =
VW
VO
2 RS =
RT
Name the orthocenter of !RST.
y
4
5
0
x
! -1 , 4 "
Helping You Remember
A good way to remember something is to explain
it to someone else. Suppose that a classmate is having trouble remembering whether the
center of gravity of a triangle is the orthocenter, the centroid, the incenter, or the
circumcenter of the triangle. Suggest a way to remember which point it is.
Sample answer: Associate the term centroid with center of gravity.
—————————————————————————————————————————
The centroid is the point of balance for any triangle.
—————————————————————————————————————————
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 5
80
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
NAME
DATE
5-3
PERIOD
Inequalities in One Triangle
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in the lesson.
1.
Sample
answer: how to recognize and apply
____________________________________________________
properties
of inequalities to the measures of the
____________________________________________________
angles
of a triangle
____________________________________________________
2.
Sample
answer: how to recognize and apply
____________________________________________________
properties
of inequalities to the relationships
____________________________________________________
between
the angles and sides of a triangle
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
perpendicular bisector
orthocenter
the place where three or more intersecting lines meet
the point of concurrency of the medians of a triangle
altitude
a segment of a triangle from a vertex that is perpendicular
to the line containing the opposite of the triangle
incenter
the point of concurrency of the lines containing the altitudes
of a triangle
point of concurrency
median
circumcenter
concurrent lines
centroid
Chapter 5
Review Vocabulary Match the term with its definition by
drawing a line to connect the two. (Lessons 5-1 and 5-2)
a segment with endpoints being a vertex of a triangle and
the midpoint of the opposite side
a segment, line, or plane that intersect a segment at its
midpoint and is perpendicular to the segment
three or more lines that intersect at a common point
the point of intersection of the perpendicular bisectors of the
three sides of a triangle
the point of intersection of the angle bisectors of the three
angles of a triangle
81
Glencoe Geometry
Lesson 5-3
Active Vocabulary
NAME
DATE
PERIOD
Lesson 5-3 (continued)
Main Idea
Angle Inequalities
pp. 342–343
Details
Describe the Exterior Angle Inequality Theorem in
your own words. Then use the figure below to write
two inequalities.
"
$
1
#
Sample
answer: The measure of an exterior angle of
___________________________________________________
a
triangle is greater than the measure of either remote
___________________________________________________
interior angle; m∠1 < m∠A, m∠1 < m∠B
___________________________________________________
Angle-Side Inequalities
pp. 343–345
List the angles of "LMN in
order from least to greatest.
/
11.6
7.5
-
10.2
.
Helping You Remember
Explain how the Exterior Angle Inequality
Theorem is related to the Exterior Angle Theorem, and why the Exterior Angle
Inequality Theorem must be true if the Exterior Angle Theorem is true.
Sample answer: The Ext. Angle Theorem says that the measure of an exterior
—————————————————————————————————————————
angle of a triangle is equal to the sum of the measures of the two remote
—————————————————————————————————————————
interior angles, while the Ext. Angle Inequality Theorem says that the measure
—————————————————————————————————————————
of an exterior angle is greater than the measure of either remote interior angle.
—————————————————————————————————————————
If
a number is equal to the sum of two positive numbers, it must be greater
—————————————————————————————————————————
than
each of those two numbers.
—————————————————————————————————————————
Chapter 5
82
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
∠L, ∠N, ∠M
NAME
5-4
DATE
PERIOD
Indirect Proof
What You’ll Learn
Scan Lesson 5-4. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: Indirect Algebraic Proof
____________________________________________________
____________________________________________________
2.
Sample
answer: Indirect Proof with Geometry
____________________________________________________
____________________________________________________
indirect reasoning
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
indirect proof
proof by contradiction
New Vocabulary Fill in each blank with the correct term or
phrase.
Indirect reasoning is a method of thinking that assumes
false
and then
that a conclusion is
showing that this assumption leads to a contradiction.
In an indirect proof, you temporarily assume that what you
are trying to prove is false. By showing this
assumption to be logically impossible, you prove your
assumption false and the original conclusion true.
To construct a proof by contradiction, the first step is to
assume
that the conclusion you want to
prove is false.
Vocabulary Link Think of other times you have encountered
the word indirect in mathematics. Describe any similarities
to an indirect proof.
Sample answer: indirect measurement; You don’t
___________________________________________________
directly measure a quantity just as you don’t directly
___________________________________________________
prove
a conclusion.
___________________________________________________
___________________________________________________
Chapter 5
83
Glencoe Geometry
Lesson 5-4
Active Vocabulary
NAME
DATE
PERIOD
Lesson 5-4 (continued)
Main Idea
Indirect Algebraic Proof
pp. 351–353
Details
Complete the table below showing the steps involved
in constructing an indirect proof.
How to Write an Indirect Proof
Sample answer: Identify the conclusion you
are asked to prove. Make the assumption
that this conclusion is false by assuming
that the opposite is true.
Step 2
Sample answer: Use logical reasoning to
show that this assumption leads to a
contradiction of the hypothesis, or some
other fact, such as a definition, postulate,
theorem, or corollary.
Step 3
Sample answer: Point out that because the
assumption leads to a contradiction, the
original conclusion, what you were asked to
prove, must be true.
Suppose you want to prove that the sum of interior
angles of a triangle is equal to 180º. What assumption
would you make to form an indirect proof of this
statement?
3
Assume: m∠1 + m∠2 + m∠3 ≠ 180
1
2
Helping You Remember
A good way to remember a new concept in
mathematics is to relate it to something you have already learned. How is the process of
indirect proof related to the relationship between a conditional statement and its
contrapositive?
Sample answer: The contrapositive of a statement p → q is the statement
—————————————————————————————————————
~q → ~p. In an indirect proof of a conditional statement p → q, you assume
—————————————————————————————————————
that q is false and show that this implies that p is false. That is, you show that
—————————————————————————————————————
~q → ~p is true, which is logically equivalent to p → q.
—————————————————————————————————————
Chapter 5
84
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Indirect Proof with
Geometry
pp. 353–354
Step 1
NAME
5-5
DATE
PERIOD
The Triangle Inequality
What You’ll Learn
Scan the text in Lesson 5-5. Write two facts you learned
about the triangle inequality.
1.
Sample
answer: You can use the triangle
____________________________________________________
inequality
to identify possible triangles given their
____________________________________________________
side
lengths.
____________________________________________________
2.
Sample
answer: The sum of two sides of a triangle
____________________________________________________
must
be greater than the length of the third side.
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
Review Vocabulary Label each figure with the correct term.
(Lessons 5-1 and 5-2)
"
QFSQFOEJDVMBSCJTFDUPS
#
DFOUSPJE
$
%
3
4
3
4
5
5
Lesson 5-5
BMUJUVEF
Chapter 5
85
Glencoe Geometry
NAME
DATE
PERIOD
Lesson 5-5 (continued)
Main Idea
The Triangle Inequality
pp. 360–361
Details
If the measures of two sides of a triangle are
4 centimeters and 9 centimeters, what is the least
possible whole number measure for the third side?
Step 1: Use the triangle inequality to write three
inequalities for a triangle with sides 4, 9, and x
centimeters.
4 +9 >x
9 +x >4
4 +x >9
Step 2: Solve each the inequalities.
9 +x >4
x >-5
4 +x >9
x>5
4 +9 >x
x < 13
Step 3: Use the inequalities to solve the problem.
x = 6 centimeters
Complete the two-column proof.
−− −−
Given: FI " FJ
Prove: FI + FH > HJ
Statements
−− −−
1. FI ! FJ
*
'
Reasons
1. Given
2. FI = FJ
2. Def of cong. seg.
3. FJ + FH > HJ
3. Triangle Ineq.
4. FI + FH > HJ
4. Substitution
+
)
Helping You Remember
A good way to remember a new theorem is to
state it informally in different words. How could you restate the Triangle Inequality
Theorem?
Sample answer: The sides that connect one vertex of a triangle to another is
—————————————————————————————————————————
a shorter path between the two vertices than the path that goes through the
—————————————————————————————————————————
third vertex.
—————————————————————————————————————————
Chapter 5
86
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Proofs Using the
Triangle Inequality
p. 362
NAME
5-6
DATE
PERIOD
Inequalities in Two Triangles
What You’ll Learn
Skim the lesson. Write two things you already know about
inequalities in two triangles.
1.
Sample
answer: The Hinge Theorem is also called
____________________________________________________
____________________________________________________
2.
Sample
answer: You can prove triangle
____________________________________________________
relationships
using the Hinge Theorem.
____________________________________________________
____________________________________________________
Active Vocabulary
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
altitude
perpendicular bisector
circumcenter
centroid
Review Vocabulary Write the correct term next to each
definition. (Lessons 5-1 and 5-2)
a segment of a triangle from a vertex that is
perpendicular to the line containing the opposite of the
triangle
a segment, line, or plane that intersect a segment at its
midpoint and is perpendicular to the segment
the point of intersection of the perpendicular bisectors of the
three sides of a triangle
the point of concurrency of the medians of a triangle
Vocabulary Link Describe how the hinge of a door can be
used to illustrate the Hinge Theorem.
Sample answer: The width of the door frame and the
___________________________________________________
door itself form the two sides. If you open two of the
___________________________________________________
same size doors at different angles, the door that is
___________________________________________________
open to a greater angle is farther from the door frame.
___________________________________________________
Chapter 5
87
Glencoe Geometry
Lesson 5-6
the
SAS Inequality Theorem.
____________________________________________________
NAME
DATE
PERIOD
Lesson 5-6 (continued)
Main Idea
Hinge Theorem
pp. 367–370
Details
Find the range of possible values for x in the figure
below.
14
12
10
61°
(3x - 5)°
10
Write an inequality
using the Converse of
the Hinge Theorem.
Write an inequality
given the angle is less
than 180º.
Solve for x.
Solve for x.
2
x > 61−
3
61 < 3x - 5
3x - 5 < 180
x > 22
What are the possible values of x ?
2
22 < x < 61−
3
Describe how you could use the
Hinge Theorem to complete the
proof below.
Given: AC = CD
Prove: BD > AB
"
#
%
$
Sample answer: You know that 2 sides of "BDC are
congruent to 2 sides of "ABC. Since the included
angle of "BDC is greater, the Hinge Theorem says
that the third side is greater than the third side of
"ABC. So, BD > AB. The distance between the points
increases or decreases accordingly.
Helping You Remember
A good way to remember something is to think of
it in concrete terms. How can you illustrate the Hinge Theorem with everyday objects?
Sample answer: Put two pencils on a desktop so that the erasers touch.
—————————————————————————————————————————
As you increase or decrease the measure of the angle formed by the pencils,
—————————————————————————————————————————
the distance between the points increases of decreases accordingly.
—————————————————————————————————————————
Chapter 5
88
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Prove Relationships in
Two Triangles
pp. 370–371
NAME
DATE
PERIOD
Relationships in Triangles
Tie It Together
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Complete each graphic organizer with a term, description, diagram, or theorem from the
chapter.
Point
of
Concurrency
Property
of
Point
Segment
Definition
perpendicular
bisector
a segment that
bisects the side
of a triangle at
a right angle
circumcenter
center of the
circumscribed
circle
angle bisector
a segment
that bisects
the angles of
a triangle
incenter
center of an
inscribed
circle
median
a segment
that connects
a vertex of a
triangle with
the midpoint
of the
opposite side
a segment
from the
vertex of a
triangle that
intersects the
opposite side
at a right
angle
centroid
center of gravity
of the triangle
orthocenter
the point of
concurrency
of the
altitudes of a
triangle
altitude
Diagram
Triangle Inequalities
Exterior Angles
Angle-Side
Triangle
Inequality
Hinge
Chapter 5
The measure of an exterior angle is greater than the measure
of either remote interior angle.
The largest angle is across from the longest side. The
smallest angle is across from the shortest side.
The longest side is across from the largest angle. The shortest
side is across from the smallest angle.
If two pairs of sides in two triangles are congruent, and the
included angle of one is larger than the included angle of the
other, then the third side of the first triangle is longer than the
third side of the second triangle.
89
Glencoe Geometry
NAME
DATE
PERIOD
Relationships in Triangles
Before the Test
Now that you have read and worked through the chapter, think about what you have
learned and complete the table below. Compare your previous answers with these.
1. Write an A if you agree with the statement.
2. Write a D if you disagree with the statement.
Relationships in Triangles
After You Read
• Concurrent lines do not intersect and stay the same
distance apart.
D
• To find the incenter of a triangle, draw a circle within
the triangle.
A
• Every triangle has 3 medians that are concurrent.
A
• The largest angle in a triangle is opposite the shortest
side.
D
• The Hinge Theorem is a way of proving triangle
relationships.
A
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 5 Study Guide and Review in the textbook.
□ I took the Chapter 5 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• Get a good nights rest before a test. Students that take the time to sleep usually do
better than students who stay up late cramming.
Chapter 5
90
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 5.
NAME
DATE
PERIOD
Quadrilaterals
Before You Read
Before you read the chapter, respond to these statements.
1. Write an A if you agree with the statement.
2. Write a D if you disagree with the statement.
Before You Read
Quadrilaterals
See students’ work.
• The sum of the interior angles of a
convex polygon is (n – 1) × 180.
• A diagonal cuts a parallelogram into
2 congruent triangles.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
• A rhombus is a quadrilateral.
• If a parallelogram has 1 right angle,
then it has 4 right angles.
• A trapezoid can have 0 or 1 set of
parallel sides.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• When you take notes, look for written real-world examples in your everyday
life.
Comment on how writers use statistics to prove or disprove points of view and discuss
the ethical responsibilities writers have when using statistics.
• When you take notes, include visuals.
Clearly label the visuals and write captions when needed.
Chapter 6
91
Glencoe Geometry
NAME
DATE
PERIOD
Quadrilaterals
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on rectangles, one fact might be a parallelogram with four right
angles is a rectangle. After completing the chapter, you can use this table to review for your
chapter test.
Lesson
6-1 Angles of Polygons
Fact
See students’ work.
6-2 Parallelograms
6-3 Tests for Parallelograms
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6-4 Rectangles
6-5 Rhombi and Squares
6-6 Trapezoids and Kites
Chapter 6
92
Glencoe Geometry
NAME
6-1
DATE
PERIOD
Angles of Polygons
What You’ll Learn
Skim Lesson 6-1. Predict two things that you expect to learn
based on the headings and figures in the lesson.
1.
Sample
answer: how to find the sum of the
____________________________________________________
____________________________________________________
2.
Sample
answer: how to find the number of sides
____________________________________________________
of
a polygon given its interior angle measures
____________________________________________________
____________________________________________________
Active Vocabulary
diagonal
New Vocabulary Write the definition next to the term.
a segment in a polygon that connects any two
————————————————————————————
nonconsecutive vertices
————————————————————————————
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Name all of the diagonals in polygon ABCDE.
"
&
#
%
−− −− −− −− −−
AC, AD, BD, BE, CE
$
Vocabulary Link Look up how television screens are
measured. What does it mean for a television to have a
32-inch screen?
Sample answer: The screens are measured along the
————————————————————————————
diagonal; 32-inch diagonal.
————————————————————————————
————————————————————————————
Chapter 6
93
Glencoe Geometry
Lesson 6-1
interior
angles of a polygon
____________________________________________________
NAME
DATE
PERIOD
Lesson 6-1 (continued)
Main Idea
Polygon Interior
Angles Sum
pp. 389–392
Details
Complete the following table for convex polygons. For
middle column, find the number of triangles you can
divide the polygon into by drawing all the possible
diagonals from one vertex.
number
of sides
number of
triangles
sum of interior
angle measures
triangle
3
1
180°
hexagon
6
4
720°
octagon
8
6
1080°
nonagon
9
7
1260°
n-gon
n
n-2
(n - 2)180°
polygon
Polygon Exterior
Angles Sum
pp. 392–393
Find the value of x in the figure below.
65°
(8x + 5)°
x = 15
(6x)°
Helping You Remember
A good way to remember a new mathematical
idea or formula is to relate it to something you already know. How can you use your
knowledge of the Angle Sum Theorem (for a triangle) to help you remember the Interior
Angle Sum Theorem?
Sample answer: The sum of the measures of the interior angles of a triangle
—————————————————————————————————————————
is 180°. Each time another side is added, the polygon can be subdivided into
—————————————————————————————————————————
one more triangle, so 180° is added to the interior angle sum.
—————————————————————————————————————————
Chapter 6
94
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
80°
NAME
6-2
DATE
PERIOD
Parallelograms
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in the lesson.
1.
Sample
answer: about the properties of
____________________________________________________
parallelograms
____________________________________________________
____________________________________________________
2.
Sample
answer: about the properties of the
____________________________________________________
diagonals
of parallelograms
____________________________________________________
Active Vocabulary
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
parallelogram
Lesson 6-2
____________________________________________________
New Vocabulary Write the definition of the term.
a quadrilateral with both pairs of opposite sides
————————————————————————————
parallel
————————————————————————————
————————————————————————————
Fill in each blank using parallelogram ABCD.
Chapter 6
−−
AB ||
−−
CD
−−−
BE "
−−
DE
−−−
AD "
−−
BC
−−−
BC ||
−−
AD
−−
AE "
−−
EC
#
"
&
%
95
$
Glencoe Geometry
NAME
DATE
PERIOD
Lesson 6-2 (continued)
Main Idea
Sides and Angles of
Parallelograms
pp. 399–400
Details
Complete the table using Theorems 6.3, 6.4, 6.5,
and 6.6 in the Student Edition.
Properties of Parallelograms
Theorem
Property
6.3
If a quadrilateral is a parallelogram, then its
opposite sides are
6.4
congruent
.
If a quadrilateral is a parallelogram, then its
opposite angles are congruent
6.5
.
If a quadrilateral is a parallelogram, then its
opposite sides are supplementary.
6.6
If a parallelogram has one right angle, then it
has four
.
Find the value of z in the parallelogram below.
9z - 4
4z + 21
z=5
Helping You Remember
A good way to remember new theorems in
geometry is to relate them to theorems you learned earlier. Name a theorem about
parallel lines that can be used to remember the theorem that says, “If a parallelogram
has one right angle, it has four right angles.”
Perpendicular Transversal Theorem
—————————————————————————————————————————
Chapter 6
96
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Diagonals of
Parallelograms
pp. 401–402
right angles
NAME
6-3
DATE
PERIOD
Tests for Parallelograms
What You’ll Learn
Skim the Examples for Lesson 6-3. Predict two things you
think you will learn about tests for parallelograms.
1.
Sample
answer: how to recognize the conditions
____________________________________________________
that
ensure a quadrilateral is a parallelogram
____________________________________________________
____________________________________________________
2.
Sample
answer: how to prove that a set of points
____________________________________________________
forms
a parallelogram in the coordinate plane
____________________________________________________
____________________________________________________
diagonal
Review Vocabulary Fill in each blank with the correct term or
phrase. (Lessons 6-1 and 6-2)
A diagonal is a segment in a polygon that connects any two
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
nonconsecutive
parallelogram
vertices.
A parallelogram is a quadrilateral with both pairs of
opposite sides
parallel.
Fill in each blank to review the properties of parallelograms.
opposite
The
congruent.
sides of a parallelogram are
The opposite angles of a parallelogram are
congruent .
Consecutive angles in a parallelogram are supplementary.
If a parallelogram has one
has four
Chapter 6
right
97
right
angle, then it
angles.
Glencoe Geometry
Lesson 6-3
Active Vocabulary
NAME
DATE
PERIOD
Lesson 6-3 (continued)
Main Idea
Conditions for
Parallelograms
pp. 409–411
Details
Complete the table using Theorems 6.9, 6.10, 6.11, and
6.12 in the student book for any quadrilateral ABCD.
Properties of Parallelograms
Theorem
6.9
6.10
6.11
Property
If both pairs of opposite sides of ABCD are
congruent , then ABCD is a parallelogram.
If both pairs of opposite angles of ABCD are
congruent , then ABCD is a parallelogram.
bisect
If the diagonals of ABCD
each other, then ABCD is a parallelogram.
If one pair of opposite sides of ABCD is both
6.12
parallel and congruent , then ABCD is a
parallelogram.
Find the slope of each line segment to verify that
WXYZ is a parallelogram.
1
−
−−−
8
slope of WX = _________
y
9
−−
5
slope of XY = _________
8
x
0
:
;
1
−
−−
8
slope of YZ = _________
−−−
5
slope of WZ = _________
Helping You Remember
A good way to remember a large number of
mathematical ideas is to think of them in groups. How can you state the conditions as
one group about the sides of a quadrilateral that guarantee it is a parallelogram?
Sample answer: both pairs of opposite sides parallel, both pairs of opposite
—————————————————————————————————————————
sides congruent, or one pair of opposite sides both parallel and congruent
—————————————————————————————————————————
Chapter 6
98
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Parallelograms on the
Coordinate Plane
pp. 412–413
NAME
6-4
DATE
PERIOD
Rectangles
What You’ll Learn
Scan Lesson 6-4. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: Properties of Rectangles
____________________________________________________
____________________________________________________
2.
Sample
answer: Prove that Parallelograms are
____________________________________________________
Rectangles
____________________________________________________
Active Vocabulary
rectangle
New Vocabulary Write the definition of the term.
a parallelogram with four right angles
————————————————————————————
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
————————————————————————————
Fill in each blank using rectangle ABCD.
"
−−
−−
BD
AC # __________
90
m∠ADC = __________
#
&
%
$
90
m∠BCD = __________
−−
−−−
BC
AD # __________
Chapter 6
99
Glencoe Geometry
Lesson 6-4
−−
−−
CD
AB "" __________
NAME
DATE
PERIOD
Lesson 6-4 (continued)
Main Idea
Details
Properties of Rectangles
pp. 419–420
Cross out the incorrect quadrilateral to complete the
Venn diagram and illustrate the relationship between
rectangles and parallelograms. Then write the
definition of the correct quadrilateral in the space
provided.
rectangle
parallelogram
rectangle
parallelogram
BRVBESJMBUFSBMXJUI
CPUIQBJSTPGPQQPTJUF
TJEFTQBSBMMFM
Use the Distance Formula to determine whether or
not parallelogram RSTU is a rectangle.
y
4
10.2
RT ≈ _________
3
5
0
x
10
SU ≈ _________
6
RSTU is not a rectangle.
Helping You Remember
It is easier to remember a large number of
geometric relationships and theorems if you are able to combine some of them. How can
you combine the two theorems about diagonals that you studied in this lesson?
Sample answer: A parallelogram is a rectangle if and only if its diagonals are
—————————————————————————————————————————
congruent.
—————————————————————————————————————————
Chapter 6
100
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Prove That
Parallelograms Are
Rectangles
pp. 420–421
BQBSBMMFMPHSBNXJUI
GPVSSJHIUBOHMFT
NAME
6-5
DATE
PERIOD
Rhombi and Squares
What You’ll Learn
Scan the text in Lesson 6-5. Write two facts you learned
about rhombi and squares.
1.
Sample
answer: The diagonals of a rhombus are
____________________________________________________
perpendicular.
____________________________________________________
____________________________________________________
2.
Sample
answer: If a quadrilateral is both a
____________________________________________________
rectangle
and a rhombus, then it is a square.
____________________________________________________
____________________________________________________
Active Vocabulary
New Vocabulary Label the diagrams with the correct terms.
rhombus
rhombus and
square
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
square
rhombus
Vocabulary Link A square is a shape that you learn at a very
early age. Name some everyday items that are shaped like a
square. Can you name any items that are shaped like a
rhombus?
Sample answer for square: tile on a floor
————————————————————————————
Sample answer for rhombus: the diamonds in a
————————————————————————————
deck
of cards
————————————————————————————
Chapter 6
101
Glencoe Geometry
Lesson 6-5
————————————————————————————
NAME
DATE
PERIOD
Lesson 6-5 (continued)
Main Idea
Properties of Rhombi
and Squares
pp. 426–428
Details
Use the properties of rhombi to solve for x in
rhombus ABCD.
AD = 8x - 11
"
#
DC = 5x + 13
&
%
$
What property of rhombi can you use to
help you solve for x?
"MMTJEFTBSFDPOHSVFOU Use the property to write an equation.
Y–=Y+
Check
Y=
Prove That
Quadrilaterals Are
Rhombi or Squares
pp. 428–430
Describe how you could prove that RS2 = RV2 + SV2 in
rhombus RSTU.
3
4
right angle because the
7
6
Sample answer: ∠RVS is a
5
diagonals are perpendicular.
Use the Pythagoren Theorem.
Chapter 6
102
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve for x.
NAME
6-6
DATE
PERIOD
Trapezoids and Kites
What You’ll Learn
Skim the lesson. Write two things you already know about
trapezoids and kites.
1.
Sample
answer: If a quadrilateral has only one set
____________________________________________________
____________________________________________________
____________________________________________________
2.
Sample
answer: The diagonals of a kite are
____________________________________________________
perpendicular
to each other.
____________________________________________________
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
New Vocabulary Match the term with its definition by
drawing a line to connect the two.
base angles
bases
isosceles trapezoid
kite
the segment that connects the midpoints of the legs of the
trapezoid
a quadrilateral with exactly two pairs of consecutive
congruent sides
the nonparallel sides of a trapezoid
the parallel sides of a trapezoid
legs of a trapezoid
the angles formed by the base and one of the legs of a
trapezoid
midsegment of a trapezoid
a quadrilateral with exactly one pair of parallel sides
trapezoid
Chapter 6
a trapezoid with congruent legs
103
Glencoe Geometry
Lesson 6-6
of
parallel sides, then it is a trapezoid.
____________________________________________________
NAME
DATE
PERIOD
Lesson 6-6 (continued)
Details
Main Idea
"
Properties of Trapezoids
pp. 435–438
#
Complete the flow proof below.
Given: ABCD is an isosceles trapezoid
−−
−−−
with bases AB and CD.
Prove: ∠BDC # ACD
%
$
−−
−−
ABCD is an isosceles trapezoid with bases AB and CD.
Given
−− −−
AC " BD
−− −−
AB " AB
Reflexive Prop.
−− −−
AD " BC
Diagonals are cong.
#BDC " #ACD
Properties of Kites
pp. 438–439
Def. isoc. trap.
∠BDC " ∠ACD
SSS
CPCTC
−−−
Solve for x if MN is a midsegment of trapezoid RSTU.
.
x=
14.4
4
16.5
/
18.6
6
x
5
Helping You Remember
A good way to remember a new geometric
theorem is to relate it to one you already know. Name and state in words a theorem
about triangles that is similar to the theorem in this lesson about the median of a
trapezoid.
Sample answer: Triangle Midsegment Theorem; A midsegment of a triangle is
—————————————————————————————————————————
parallel to one side of the triangle, and its length is one half the length of that
—————————————————————————————————————————
side.
—————————————————————————————————————————
Chapter 6
104
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
NAME
DATE
PERIOD
Quadrilaterals
Tie It Together
Complete the graphic organizer with properties of the figure from the chapter.
Quadrilaterals
Parallelograms
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
One pair of opposite sides are both
congruent and parallel.
Rectangles
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
All angles are
right angles.
Diagonals are
congruent.
Rhombi
Squares All sides are
congruent.
Diagonals are
perpendicular.
Trapezoids
Kites
One pair of sides is parallel.
Two pairs of consecutive
sides are congruent.
Diagonals are
perpendicular.
One pair of opposite
angles are congruent.
Isosceles
Trapezoids
One pair of sides is
congruent.
Both pairs of base
angles are congruent.
Diagonals are
congruent.
Chapter 6
105
Glencoe Geometry
NAME
DATE
PERIOD
Quadrilaterals
Before the Test
Now that you have read and worked through the chapter, think about what you have
learned and complete the table below. Compare your previous answers with these.
1. Write an A if you agree with the statement.
2. Write a D if you disagree with the statement.
Quadrilaterals
After You Read
• The sum of the interior angles of a convex polygon is
(n – 1) × 180.
D
• A diagonal cuts a parallelogram into 2 congruent
triangles.
A
A
• A rhombus is a quadrilateral.
• If a parallelogram has 1 right angle, then it has 4 right
angles.
• A trapezoid can have 0 or 1 set of parallel sides.
A
D
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 6 Study Guide and Review in the textbook.
□ I took the Chapter 6 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• When you are preparing to read new material, scan the text first, briefly looking over
headings, highlighted text, pictures, and call out boxes. Think of questions that you can
search the text for as you read.
Chapter 6
106
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 6.
NAME
DATE
PERIOD
Proportions and Similarity
Before You Read
Before you read the chapter, think about what you know about proportions and similarity.
List three things you already know about proportions and similarity in the first column.
Then list three things you would like to learn about them in the second column.
K
W
What I know…
What I want to find out…
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
See students’ work.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• When you take notes, write instructions on how to do something presented in
each lesson.
Then follow your own instructions to check them for accuracy.
• When you take notes, be sure to describe steps in detail.
Include examples of questions you might ask yourself during problem solving.
Chapter 7
107
Glencoe Geometry
NAME
DATE
PERIOD
Proportions and Similarity
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on similar polygons, one fact might be similar polygons have the
same shape but not the same size. After completing the chapter, you can use this table to
review for your chapter test.
Lesson
7-1 Ratios and Proportions
Fact
See students’ work.
7-2 Similar Polygons
7-3 Similar Triangles
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7-4 Parallel Lines and Proportional Parts
7-5 Parts of Similar Triangles
7-6 Similarity Transformations
7-7 Scale Drawings and Models
Chapter 7
108
Glencoe Geometry
NAME
7-1
DATE
PERIOD
Ratios and Proportions
What You’ll Learn
Skim the Examples for Lesson 7-1. Predict two things you
think you will learn about ratios and proportions.
1.
Sample
answer: how to write and use ratios
____________________________________________________
____________________________________________________
2.
Sample
answer: how to use cross products to
____________________________________________________
make
predictions
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
New Vocabulary Match the term with its definition by
drawing a line to connect the two.
cross product
a
c
the numbers a and d in the proportion −
=−
extended ratio
an equation stating that two ratios are equal
extremes
a ratio that compares three or more numbers
means
b
d
the product of the extremes, ad, and the product of the
a
c
=−
means, bc in the proportion −
b
d
proportion
a comparison of two quantities using division
ratio
a
c
=−
the numbers b and c in the proportion −
b
d
Vocabulary Link Look up the meaning of the word
proportional. How does this relate to a proportion in
mathematics?
Sample answer: properly related in size or scale
————————————————————————————
————————————————————————————
————————————————————————————
Chapter 7
109
Glencoe Geometry
Lesson 7-1
____________________________________________________
NAME
DATE
PERIOD
Lesson 7-1 (continued)
Main Idea
Write and Use Ratios
pp. 457–458
Details
The ratio of the measures of the angles in !DEF is
4:5:9. Find the measures of the angles.
Step 1: Multiply the ratio
by x to represent the
unknown angle measures.
Use Properties of
Proportions
pp. 458–459
4x : _______
9x
5x : _______
_______
Step 2: Write an equation
using the extended ratio and
the Triangle Sum Theorem.
_________________________
Step 3: Solve for x.
x = 10
_________________________
Step 4: Find the angle
measures.
40°, 50°, 90°
_________________________
4x + 5x + 9x = 180
3
x
=−
−
Write a proportion.
8000
120
_________________________
Solve the proportion.
200 stereos
_________________________
Helping You Remember
Sometimes it is easier to remember a
mathematical idea if you put it into words without using any mathematical symbols.
How can you use this approach to remember the concept of equality of cross products?
Sample answer: The product of the means equals the product of the extremes.
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 7
110
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A quality control engineer randomly samples 120 car
stereos coming off an assembly line. He finds that 3 of
them have cosmetic blemishes. How many car stereos
would you expect to have cosmetic blemishes in a
production run of 8000 car stereos?
NAME
DATE
7-2
PERIOD
Similar Polygons
What You’ll Learn
Skim Lesson 7-2. Predict two things that you expect to learn
based on the headings and figures in the lesson.
1.
Sample
answer: how to identify similar figures
____________________________________________________
____________________________________________________
____________________________________________________
2.
Sample
answer: how to use similar figures to find
____________________________________________________
missing
measures
____________________________________________________
Active Vocabulary
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
proportion
extended ratio
extremes
ratio
Review Vocabulary Write the definition next to each term.
(Lesson 7-1)
an equation stating that two ratios are equal
————————————————————————————
a ratio that compares three or more numbers
————————————————————————————
c
a
−
the numbers b and c in the proportion −
c = d
————————————————————————————
a comparison of two quantities using division
————————————————————————————
New Vocabulary Write the correct term next to each
definition.
similarity ratio
similar polygons
scale factor
Chapter 7
the scale factor between two similar polygons
polygons that have the same shape but not necessarily the
same size
the ratio of the lengths of the corresponding sides of two
similar polygons
111
Glencoe Geometry
Lesson 7-2
____________________________________________________
NAME
DATE
PERIOD
Lesson 7-2 (continued)
Main Idea
Identify Similar
Polygons
pp. 465–466
Details
Model two similar triangles by writing the side
lengths of !XYZ. Then tell the scale factor from
!HIJ to !XYZ.
Sample answer is given.
)
9
20
16
16
12.8
;
+
12
*
:
9.6
1.25
scale factor: _____________
Use Similar Figures
pp. 467–468
Solve for y in the similar trapezoids below.
27
18
2.5
y = _____________
Helping You Remember
A good way to remember a new mathematical
vocabulary term is to relate it to words used in everyday life. The word scale has many
meanings in English. Give three phrases that include the word scale in a way that is
related to proportions.
Sample answer: scale drawing, scale model, scale of miles (on a map)
—————————————————————————————————————————
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 7
112
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2y + 1
9
NAME
7-3
DATE
PERIOD
Similar Triangles
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in the lesson.
1.
Sample
answer: how to identify similar triangles
____________________________________________________
using
the AA Postulate, SSS Theorem, and SAS
____________________________________________________
Theorem
____________________________________________________
____________________________________________________
2.
Sample
answer: how to use similar triangles to
____________________________________________________
solve
problems
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
extremes
Review Vocabulary Fill in each blank with the correct term or
phrase. (Lessons 7-1, 7-2)
a
d
The extremes are the numbers ___________
and ___________
a
c
=−
.
in the proportion −
b
means
b
c
The means are the numbers ___________
and ___________
in
a
c
=−
.
the proportion −
b
proportion
d
d
two ratios
A proportion is an equation stating that __________________
are equal.
scale factor
The scale factor is the ratio of the lengths of the
corresponding sides of two similar polygons.
____________________
ratio
similar polygons
division
A ratio is a comparison of two quantities using ____________.
shape but not
Similar polygons have the same ____________
size
necessarily the same ____________.
Chapter 7
113
Glencoe Geometry
Lesson 7-3
____________________________________________________
NAME
DATE
PERIOD
Lesson 7-3 (continued)
Main Idea
Identify Similar
Triangles
pp. 474–477
Details
Determine whether the triangles are similar. If so,
write a similarity statement. Explain your reasoning.
5
0
77°
7
65°
3
65°
77°
4
:
!ORS
∼ !YVT by AA similarity.
____________________________________________________
Use Similar Triangles
pp. 477–478
Complete the table to show the properties of
similarity in Theorem 7.5.
Theorem 7.5
Properties of Similarity
!ABC ∼ !ABC
Symmetric Property
If !ABC ∼ !DEF, then
!DEF ∼ !ABC.
Transitive Property
If !ABC ∼ !DEF, and !DEF ∼
!LMN, then !ABC ∼ !LMN.
Helping You Remember
A good way to remember something is to explain
it to somone else. Suppose one of your classmates is having trouble understanding the
difference between SAS for congruent triangles and SAS for similar triangles. How can
you explain the difference to him or her?
Sample answer: In SAS for congruence, corresponding sides are congruent,
—————————————————————————————————————————
while in SAS for similarity, corresponding sides are proportional. In both, the
—————————————————————————————————————————
included angles are congruent.
—————————————————————————————————————————
Chapter 7
114
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Reflexive Property
NAME
7-4
DATE
PERIOD
Parallel Lines and Proportional Parts
What You’ll Learn
Scan Lesson 7-4. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: Proportional Parts Within
____________________________________________________
Triangles
____________________________________________________
2.
Sample
answer: Proportional Parts with Parallel
____________________________________________________
Lines
____________________________________________________
Active Vocabulary
extreme
mean
extended ratio
Review Vocabulary Label the diagrams with the correct
terms. (Lesson 7-1)
mean
3
29
−
=−
6
58
extreme
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4 : 8 : 11
extended ratio
New Vocabulary Label the diagram with the correct terms.
.
Lesson 7-4
"
/
midsegment
#
Chapter 7
115
$
Glencoe Geometry
NAME
DATE
PERIOD
Lesson 7-4 (continued)
Main Idea
Proportional Parts
Within Triangles
pp. 484–486
Details
Use the Triangle Proportionality Theorem to find RT.
.
Step 1: Use the Triangle
Proportionality Theorem
to write a proportion.
MQ
RT
− =−
RS
QS
______________________
Step 2: Substitute the
known values into the
proportion.
3.6
RT
=−
−
8.4
6.3
______________________
Proportional Parts with
Parallel Lines
pp. 486–488
5
3.6
2
3
8.4
6.3
4
Step 3: Solve the
proportion for RT.
RT = 4.8
______________________
Model Corollary 7.1 by drawing two transversals on
the parallel lines below and writing a proportion.
Sample answers are given.
#
k
m
9
:
$
XY
AB
=−
−
;
YZ
BC
______________________
Helping You Remember
A good way to remember a new mathematical
term is to relate it to other mathematical vocabulary that you already know. What is an
easy way to remember the definition of midsegment using other geometric terms?
Sample answer: A midsegment is a segment that connects the midpoints of
—————————————————————————————————————————
two sides of a triangle.
—————————————————————————————————————————
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 7
116
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
"
l
NAME
7-5
DATE
PERIOD
Parts of Similar Triangles
What You’ll Learn
Scan the text in Lesson 7-5. Write two facts you learned
about parts of similar triangles.
1.
Sample
answer: If two triangles are similar, the
____________________________________________________
lengths
of corresponding altitudes are
____________________________________________________
proportional
to the lengths of corresponding sides.
____________________________________________________
____________________________________________________
2.
Sample
answer: You can use special segments in
____________________________________________________
similar
triangles to solve for missing measures.
____________________________________________________
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
Review Vocabulary Label the diagrams with the correct
terms. (Lessons 5-1 and 5-2)
angle bisector
altitude
altitude
median
angle bisector
median
Chapter 7
117
Glencoe Geometry
Lesson 7-5
50° 50°
NAME
DATE
PERIOD
Lesson 7-5 (continued)
Main Idea
Special Segments of
Similar Triangles
pp. 495–497
Details
Use Theorem 7.8 in the student book to solve for BD
in the similar triangles below.
#
"
4
$
16
15
3
%
Step 1: Use Theorem 7.8 to
write a proportion.
Step 2: Substitute the known
values into the proportion.
Step 3: Solve the proportion
for BD.
6
5
AD
BD
=−
−
ST
US
______________________
15
BD
=−
−
20
16
______________________
BD = 12
______________________
Model the Triangle Angle Bisector Theorem by
drawing an angle bisector and writing a proportion.
Sample answer:
6
3
4
RU
RT
=−
−
ST
SU
______________________
5
Helping You Remember
A good way to remember a large amount of
information is to remember key words. What key words will help you remember the
features of similar triangles that are proportional to the lengths of the corresponding
sides?
Sample answer: perimeters, altitudes, angle bisectors, medians
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 7
118
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Triangle Angle
Bisector Theorem
p. 498
20
NAME
7-6
DATE
PERIOD
Similarity Transformations
What You’ll Learn
Skim the lesson. Write two things you already know about
similarity transformations.
1.
Sample
answer: A similarity transformation is a
____________________________________________________
figure.
____________________________________________________
____________________________________________________
2.
Sample
answer: You can use SAS Similarity to
____________________________________________________
verify
that two figures are similar after a dilation.
____________________________________________________
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
center of dilation
dilation
New Vocabulary Fill in each blank with the correct term or
phrase.
fixed point
Dilations are performed with respect to a _______________
called the center of dilation.
enlarges
A dilation is a transformation that _______________
or
reduces
_______________
the original figure proportionally.
enlargement
scale factor of a dilation
similarity transformation
greater than 1 produces
A dilation with a scale factor ____________________
an enlargement.
extent
The scale factor of a dilation describes the _______________
of the dilation.
A similarity transformation is a transformation that maps a
similar
figure onto a _______________
figure.
reduction
Chapter 7
between 0 and 1
A dilation with a scale factor _____________________
produces a reduction.
119
Glencoe Geometry
Lesson 7-6
transformation
that maps a figure onto a similar
____________________________________________________
NAME
DATE
PERIOD
Lesson 7-6 (continued)
Main Idea
Identify Similarity
Transformations
pp. 505–506
Details
Model a reduction and an enlargement by drawing a
similar triangle on each coordinate grid. Write the
scale factor of each dilation. Sample answer:
Reduction
Enlargement
y
"
y
#
"'
"'
#'
"
#'
x
0
x
0
$'
$
$'
$
0.5
scale factor: __________
Verify Similarity
p. 507
#
1.5
scale factor: __________
Graph the original figure and its dilated image on a
sheet of graph paper. Verify that the dilation is a
similarity transformation. See students’ work.
The transformation is an enlargement with a scale
factor of 2.
Image: A!(0, 0), B!(4, 6), C!(8, 2)
Helping You Remember
A good way to develop a higher understanding of
a concept is to compare and contrast it with other similar topics. Compare and contrast
similarity transformations with the congruency transformations that you learned about
in chapter 6.
Sample answer: Both types of transformations deal with taking a geometric
—————————————————————————————————————————
figure (preimage) and mapping it onto an image. With congruency
—————————————————————————————————————————
transformations, the image is congruent to the preimage. With similarity
—————————————————————————————————————————
transformations, the image is similar to the preimage.
—————————————————————————————————————————
Chapter 7
120
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Preimage: A(0, 0), B(2, 3), C(4, 1)
NAME
7-7
DATE
PERIOD
Scale Drawings and Models
What You’ll Learn
Scan the text in Lesson 7-7. Write two facts you learned
about scale drawings and models.
1.
Sample
answer: A scale model is an object with
____________________________________________________
lengths
proportional to the object it represents.
____________________________________________________
____________________________________________________
2.
Sample
answer: Scale factors are always written
____________________________________________________
so
that the model length in the ratio comes first.
____________________________________________________
Active Vocabulary
scale
New Vocabulary Write the definition next to each term.
the ratio of a length on a scale model or drawing to
————————————————————————————
the actual length of the object being modeled or drawn
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
————————————————————————————
scale drawing
a drawing with lengths proportional to the object
————————————————————————————
that it represents
————————————————————————————
scale model
an object with lengths proportional to the object that
————————————————————————————
it represents
————————————————————————————
New Vocabulary Scale drawings and models have a wide
range of applications. For instance, a roadmap is a real
world example of a scale drawing. Can you think of other
real world examples of scale drawings and scale models?
Sample answers: blueprints, toy models, photographs
————————————————————————————
————————————————————————————
Chapter 7
121
Glencoe Geometry
Lesson 7-7
____________________________________________________
NAME
DATE
PERIOD
Lesson 7-7 (continued)
Main Idea
Details
Scale Models
pp. 512–513
The scale on a map is 2 centimeters : 15 kilometers. If
the distance between two cities on the map is 9
centimeters, find the actual distance between the
cities.
Method 1
Write a proportion
using the scale.
9 cm
2 cm
−
=−
Check
Method 2
Write an equation
using the number of
kilometers per
centimeters.
15
a= −
m
15 km
x km
______________________
2
______________________
Solve the proportion.
Solve the equation.
Substitute 9 for m.
x = 67.5 km
______________________
a = 67.5 km
______________________
Check
Helping You Remember
A good way to remember something is to explain
it to someone else. Suppose one of your classmates is having trouble understanding how
to use the scale on a map to calculate actual distances. How can you explain the
procedure to her?
Sample answer: Use a ruler to measure the distance on the map. Set up a
—————————————————————————————————————————
proportion to solve for the actual distance. Use the scale of the map (map
—————————————————————————————————————————
distance : actual distance) to write one ratio. Use the ratio measured distance :
—————————————————————————————————————————
unknown actual distance as the other. Solve.
—————————————————————————————————————————
Chapter 7
122
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Suppose a model airplane has a wingspan of 8 inches.
The actual plane has a wingspan of 48 feet.
1 in. : ______
6 ft
1. What is the scale of the model? ______
2. How many times as long as the actual plane is the
1
−
72
model? ______________
NAME
DATE
PERIOD
Proportions and Similarity
Tie It Together
Complete the graphic organizer with a phrase or formula from the chapter.
Equivalent
Proportions
a
c b
d
−
=−
,−
=−
,
a
c
b
d
a
d
b c
−
=−
,−=−
c
d a
b
Definition
transformation
in which a figure
is reduced or
enlarged
Definition
an object with
lengths
proportional to
the object it
represents
Cross Products
a
c
If −
=−
,
b
d
then ad = bc.
Dilations
Scale
Drawings
Algebra
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Definition
corresponding
angles congruent
corresponding
sides proportional
Ratio
Similar
Polygons
Theorems
• Triangle Proportionality
Similar
Triangles
• Triangle Midsegment
• Proportional Parts of Parallel Lines
• Congruent Parts of Parallel Lines
Theorems
Properties
AA
SSS
SAS
Reflexive
Symmetric
Transitive
Chapter 7
• Special Segments Proportionality
• Triangle Angle Bisector
123
Glencoe Geometry
NAME
DATE
PERIOD
Proportions and Similarity
Before the Test
Review the ideas you listed in the table at the beginning of the chapter. Cross out any
incorrect information in the first column. Then complete the table by filling in the third
column.
K
W
L
What I know…
What I want to find out…
What I learned…
See students’ work.
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 7 Study Guide and Review in the textbook.
□ I took the Chapter 7 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• Complete reading assignments before class. Write down or circle any questions you may
have about what was in the text.
Chapter 7
124
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 7.
NAME
DATE
PERIOD
Right Triangles and Trigonometry
Before You Read
Before you read the chapter, think about what you know about right triangles and
trigonometry. List three things you already know about them in the first column. Then list
three things you would like to learn about them in the second column.
K
W
What I know…
What I want to find out…
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
See students’ work.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• When searching for the main idea of a lesson, ask yourself, “What is this
paragraph or lesson telling me?”
• When you take notes, include definitions of new terms, explanations of new
concepts, and examples of problems.
Chapter 8
125
Glencoe Geometry
NAME
DATE
PERIOD
Right Triangles and Trigonometry
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on geometric mean, one fact might be the geometric mean
between two numbers is the positive square root of their product. After completing the
chapter, you can use this table to review for your chapter test.
Lesson
8-1 Geometric Mean
Fact
See students’ work.
8-2 The Pythagorean Theorem and Its
Converse
8-3 Special Right Triangles
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8-4 Trigonometry
8-5 Angles of Elevation
8-6 The Law of Sines and Law of Cosines
8-7 Vectors
Chapter 8
126
Glencoe Geometry
NAME
8-1
DATE
PERIOD
Geometric Mean
What You’ll Learn
Skim Lesson 8-1. Predict two things that you expect to learn
based on the headings and the Key Concept box.
1.
Sample
answer: how to find the geometric mean
____________________________________________________
____________________________________________________
2.
Sample
answer: how to solve problems involving
____________________________________________________
right
triangles using the geometric mean
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
Review Vocabulary Write the correct term next to each
definition. (Lesson 7-1)
proportion
an equation stating that two ratios are equal
means
a
c
the numbers a and d in the proportion − = −
extremes
a
c
the numbers b and c in the proportion −
=−
b
b
d
d
New Vocabulary Complete each statement by filling in the
blank with the correct term or phrase or writing the correct
formula.
geometric mean
The geometric mean of two positive numbers a and b is the
number x such that
x
a =−
−
x
b
.
The measure of the altitude drawn from the vertex of a right
triangle to its hypotenuse is the geometric mean between
hypotenuse
the measures of the two segments of the ________________.
The geometric mean of two positive numbers a and b can be
calculated using the expression
Chapter 8
127
x=
√##
ab
.
Glencoe Geometry
Lesson 8-1
of
two numbers
____________________________________________________
NAME
DATE
PERIOD
Lesson 8-1 (continued)
Main Idea
Geometric Mean
p. 531
Geometric Means in
Right Triangles
pp. 531–534
Details
Find the geometric mean between 8 and 18.
x = √""
ab
__________________
definition of geometric mean
8·18
x = √""
__________________
Substitute for a and b.
144
x = √""
__________________
Multiply.
x = 12
__________________
Simplify.
Model Theorem 8.2, the Geometric Mean (Altitude)
Theorem, by drawing a segment on right triangle DEF
and writing a proportion. Sample answers are given.
%
,
&
'
√""""
FK = DK · KE
_______________________
Helping You Remember
A good way to remember a new mathematical
concept is to relate it to something you already know. How can the meaning of mean in a
proportion help you to remember how to find the geometric mean between two numbers?
Sample answer: Write a proportion in which the two means are equal. This
—————————————————————————————————————————
common mean is the geometric mean between the two extremes.
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 8
128
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
90°
NAME
DATE
8-2
PERIOD
The Pythagorean Theorem and Its Converse
What You’ll Learn
Scan the text in Lesson 8-2. Write two facts you learned
about the Pythagorean Theorem and its converse as you
scanned the text.
1.
Sample
answer: In a right triangle, the sum of the
____________________________________________________
squares
of the lengths of the legs is equal to the
____________________________________________________
square
of the length of the hypotenuse.
____________________________________________________
2.
Sample
answer: The converse of the Pythagorean
____________________________________________________
whether
a triangle is a right triangle given the
____________________________________________________
measures
of all three sides.
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
geometric mean
Review Vocabulary Fill in each blank with the correct term or
phrase. (Lesson 1-5)
positive
The geometric mean of two ________________
numbers a
and b can be calculated using the expression x = √""
ab .
altitude
The measure of the ________________
drawn from the
vertex of the right triangle to its hypotenuse is the
geometric mean between the measures of the two
segments of the hypotenuse.
New Vocabulary Write the definition next to the term.
Pythagorean triple
a set of three nonzero whole numbers a, b, and c,
————————————————————————————
such that a2 + b2 = c2
————————————————————————————
————————————————————————————
Chapter 8
129
Glencoe Geometry
Lesson 8-2
Theorem
can be used to help you determine
____________________________________________________
NAME
DATE
PERIOD
Lesson 8-2 (continued)
Main Idea
The Pythagorean
Theorem
pp. 541–544
Converse of the
Pythagorean Theorem
pp. 544–545
Details
Circle each set of numbers that is a Pythagorean
triple.
1. 6, 8, 10
2. 7, 21, 25
3. 5, 12, 13
4. 24, 45, 51
5. 14, 48, 50
6. 10, 15, 17
7. 21, 72, 75
8. 16, 30, 36
9. 15, 36, 39
Fill in each blank to complete the converse of the
Pythagorean Theorem.
#
c
a
$
b
"
the shortest sides of a triangle is equal to the
longest side,
square of the length of the ____________
then the triangle is a right triangle.
Symbols
!ABC is a right triangle
If a2 + b2 = c2, then _________________________.
Helping You Remember
Many students who studied geometry long ago
remember the Pythagorean Theorem as the equation a2 + b2 = c2, but cannot tell you
what this equation means. A formula is useless if you don’t know what it means and
how to use it. How could you help someone who has forgotten the Pythagorean Theorem
remember the meaning of the equation a2 + b2 = c2?
Sample answer: Draw a right triangle. Label the lengths of the two legs as a
—————————————————————————————————————————
and b and the length of the hypotenuse as c.
—————————————————————————————————————————
Chapter 8
130
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Converse of the Pythagorean Theorem
Words
squares of the lengths of
If the sum of the ____________
NAME
8-3
DATE
PERIOD
Special Right Triangles
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in the lesson.
1.
Sample
answer: how to use the properties of
____________________________________________________
45º-45º-90º
triangles
____________________________________________________
2.
Sample
answer: how to use the properties of
____________________________________________________
30º-60º-90º
triangles
____________________________________________________
Active Vocabulary
New Vocabulary Quadrilateral ABCD is a square with side
lengths s. Fill in each blank to show what you know about
the figure.
s
AB = ________
"
#
90°
m∠ADC = ________
s
%
$
s √"
2
AC = ________
Triangle LMN is equilateral with side lengths 2s. Fill in
each blank to show what you know about the figure.
s
ON = ________
-
30°
m∠OLN = ________
2s
60°
m∠LNO = ________
.
0
/
√"
3
LO = ________
Chapter 8
131
Glencoe Geometry
Lesson 8-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
45°
m∠ACD = ________
NAME
DATE
PERIOD
Lesson 8-3 (continued)
Main Idea
Properties of 45º-45º-90º
Triangles
pp. 552–553
Details
Fill in each blank and label the figure to illustrate the
45º-45º-90º Triangle Theorem.
45º-45º-90º Triangle Theorem
5
In a 45º-45º-90º triangle, the legs ℓ
45°
! 2
congruent and the length
are ______________
√"
2
of the hypotenuse h is ___________
3
!
45°
!
4
times the length of a leg.
Properties of 30º-60º-90º
Triangles
pp. 553–555
Solve for q and s in the figure below.
2
7.5
q = ____________
15
s
60°
q
4
7.5 √"
3
s = ____________
Helping You Remember
Some students find it easier to remember
mathematical concepts in terms of specific numbers rather than variables. How can you
use specific numbers to help you remember the relationship between the lengths of the
three sides in a 30º-60º-90º triangle?
Sample answer: Draw a 30º-60º-90º triangle. Label the length of the shorter leg
—————————————————————————————————————————
as 1. Then the length of the hypotenuse is 2, and the length of the longer leg
—————————————————————————————————————————
is √"
3 . Just remember 1, 2, √"
3.
—————————————————————————————————————————
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 8
132
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
NAME
8-4
DATE
PERIOD
Trigonometry
What You’ll Learn
Skim the Examples for Lesson 8-4. Predict two things you
think you will learn about trigonometry.
1.
Sample
answer: how to find the sine, cosine, and
____________________________________________________
tangent
ratios for angles in a right triangle
____________________________________________________
____________________________________________________
2.
Sample
answer: how to use trigonometric ratios to
____________________________________________________
find
lengths
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
sine
______________________
New Vocabulary Write the correct term next to each
definition.
the ratio of the opposite leg to the hypotenuse of a right
triangle
inverse sine
______________________
the measure of ∠A if sin A is known
trigonometry
______________________
the study of triangle measurement
inverse tangent
______________________
the measure of ∠A if tan A is known
cosine
______________________
the ratio of the adjacent leg to the hypotenuse of a right
triangle
inverse cosine
______________________
the measure of ∠A if cos A is known
trigonometric ratio
______________________
a ratio of the lengths of two sides of a right triangle
tangent
______________________
the ratio of the opposite leg to the adjacent leg of a right
triangle
Chapter 8
133
Glencoe Geometry
Lesson 8-4
Active Vocabulary
NAME
DATE
PERIOD
Lesson 8-4 (continued)
Main Idea
Trigonometric Ratios
pp. 562–564
Details
Complete the statements to show the trigonometric
ratios for angles H and K.
h
−
"
sin H =
)
k
−
"
sin K =
k
k
−
"
cos H =
-
h
−
h
,
"
cos K =
h
−
k
tan H =
k
−
h
tan K =
Use a calculator to find m∠F to the nearest tenth.
'
16
&
(
9
34.2
m∠F ≈_________
Helping You Remember
How can the co in cosine help you remember the
relationship between the sines and the cosines of the two acute angles of a right
triangle?
Sample answer: The co in cosine comes from complement, as in
—————————————————————————————————————————
complementary angles. The cosine of an acute angle is equal to the sine of its
—————————————————————————————————————————
complement.
—————————————————————————————————————————
Chapter 8
134
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Use Inverse
Trigonometric Ratios
pp. 564–566
!
NAME
DATE
8-5
PERIOD
Angles of Elevation and Depression
What You’ll Learn
Skim the lesson. Write two things you already know about
angles of elevation and depression.
1.
Sample
answer: An angle of elevation is the angle
____________________________________________________
formed
by a horizontal line and an observer’s line
____________________________________________________
of
sight to an object above the horizontal line.
____________________________________________________
2.
Sample
answer: You can use inverse trigonometric
____________________________________________________
ratios
to solve for angles of elevation and
____________________________________________________
depression
if you know two of the side lengths.
____________________________________________________
Active Vocabulary
angle of elevation
New Vocabulary Write the definition next to each term.
the angle formed by a horizontal line and an
————————————————————————————
————————————————————————————
horizontal line
————————————————————————————
angle of depression
the angle formed by a horizontal line and an
————————————————————————————
observer’s line of sight to an object below the
————————————————————————————
horizontal line
————————————————————————————
Vocabulary Link What are some real-world examples of
angles of elevation and depression?
Sample answer: When a plane takes off, its climb
————————————————————————————
forms an angle of elevation. When a plane lands, it
————————————————————————————
comes in at an angle of depression.
————————————————————————————
Chapter 8
135
Glencoe Geometry
Lesson 8-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
observer’s line of sight to an object above the
NAME
DATE
PERIOD
Lesson 8-5 (continued)
Main Idea
Angles of Elevation and
Depression
pp. 574–575
Details
A plane takes off and climbs at an angle of elevation
of 35º. How high above the ground is the plane after
traveling 250 yards? Model the situation with a
diagram and solve.
Sample answer is given.
h
sin 35° = −
250
250 yd
h
250(sin 35°) = h
143.4 ≈ h
35°
about 143.4 yards
Two Angles of Elevation
or Depression
pp. 575–576
Solve for x in the figure below.
25°
20 m
20°
22.9 m
x ≈ __________
Helping You Remember
A good way to remember something is to explain
it to someone else. Suppose a classmate finds it difficult to distinguish between angles of
elevation and angles of depression. What are some hints you can give her to get it right
every time?
Sample answers: (1) The angle of depression and the angle of elevation are
—————————————————————————————————————————
both measured between the horizontal and the line of sight. (2) The angle of
—————————————————————————————————————————
depression is always congruent to the angle of elevation in the same diagram.
—————————————————————————————————————————
(3) Associate the word elevation with the word up and the word depression
—————————————————————————————————————————
with the word down.
—————————————————————————————————————————
Chapter 8
136
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x
NAME
The Law of Sines and Law of Cosines
What You’ll Learn
Active Vocabulary
tangent
angle of depression
cosine
angle of elevation
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
PERIOD
Scan Lesson 8-6. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: Law of Sines
____________________________________________________
2.
Sample
answer: Law of Cosines
____________________________________________________
Review Vocabulary Match the term with its definition by
drawing a line to connect the two. (Lesson 1-5)
the angle formed by a horizontal line and an observer’s line
of sight to an object above the horizontal line
the ratio of the opposite leg to the adjacent leg of a right
triangle
the angle formed by a horizontal line and an observer’s line
of sight to an object above the horizontal line
the ratio of the adjacent leg to the hypotenuse of a right
triangle
New Vocabulary Fill in each blank with the correct term or
phrase.
Law of Sines
triangle
You can use the Law of Sines to solve a ________________
if
two angles and
you know the measures of ________________
any side
________________
(AAS). If given ASA, use the Triangle
Angle Sum Theorem to find the measure of the
third angle
________________.
Law of Cosines
You can use the Law of Cosines to solve a triangle if you
sides
know the measures of two ________________
and the
included angle (SAS).
__________________
Chapter 8
137
Glencoe Geometry
Lesson 8-6
8-6
DATE
NAME
DATE
PERIOD
Lesson 8-6 (continued)
Main Idea
Law of Sines
pp. 582–583
Details
Complete the statement below to illustrate the Law of
Sines for !ABC.
Law of Sines
If !ABC has lengths a, b,
and c, representing the
length of the sides opposite
the angles with measures A,
B, and C, then
"
#
sin C
sin A
sin B
−
=−
=−
a
c
b
.
Law of Cosines
pp. 583–585
b
c
$
a
Use the Law of Cosines to solve for x to the
nearest tenth.
3
18
x
15
5
x ≈ 41.4°
_____________
Helping You Remember
Many students remember mathematical
equations and formulas better if they can state them in words. State the Law of Sines in
your own words without using variables or mathematical symbols.
Sample answer: In any triangle, the ratio of the sine of an angle to the length of
—————————————————————————————————————————
the opposite side is the same for all three angles.
—————————————————————————————————————————
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 8
138
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
12
NAME
8-7
DATE
PERIOD
Vectors
What You’ll Learn
Skim the Examples for Lesson 8-7. Predict two things you
think you will learn about vectors.
1.
Sample
answer: how to write vectors in
____________________________________________________
component
form
____________________________________________________
2.
Sample
answer: how to find the direction and
____________________________________________________
Active Vocabulary
component form
______________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
direction
______________________
magnitude
______________________
New Vocabulary Write the correct term next to each
definition.
a description of a vector in terms of its horizontal change x
and vertical change y from its initial point to its terminal
point
the angle that is formed with the x-axis or any other
horizontal line
the length of the vector from its initial point to its terminal
point
parallelogram method
______________________
a way to find the sum of two vectors using a parallelogram
resultant
______________________
the sum of two vectors
standard position
______________________
a vector that is placed in the coordinate plane with its initial
point at the origin
triangle method
______________________
a way to find the sum of two vectors using a triangle
vector
______________________
a quantity that has both magnitude and direction
Chapter 8
139
Glencoe Geometry
Lesson 8-7
magnitude
of a vector
____________________________________________________
NAME
DATE
PERIOD
Lesson 8-7 (continued)
Main Idea
Describe Vectors
pp. 593–594
Details
".
Write the component form of ST
y
5
0
〈7, 6〉
Vector Addition
pp. 594–596
x
4
Model the parallelogram method and the triangle
method below by drawing two vectors and showing
the resultant.
Sample answers are shown.
Parallelogram Method
Triangle Method
A good way to remember a new mathematical
term is to relate it to a term you already know. You learned about scale factors when you
studied similarity and dilations. How is the idea of a scalar related to scale factors?
Sample answer: A scalar is the term used for a constant (a specific real
—————————————————————————————————————————
number) when working with vectors. A vector has both magnitude and
—————————————————————————————————————————
direction, while a scalar is just a magnitude. Multiplying a vector by a positive
—————————————————————————————————————————
scalar changes the magnitude of the vector, but not the direction, so it
—————————————————————————————————————————
represents a change in scale.
—————————————————————————————————————————
Chapter 8
140
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Helping You Remember
NAME
DATE
PERIOD
Right Triangles and Trigonometry
Tie It Together
Complete the graphic organizer with a term or formula from the chapter.
Right Triangles
Notation
Diagram/Examples
a
x
−
−
x =
3
x
−
−
; x2 = 45; x = 3 √#
5
x =
Similar Right Triangles
Geometric Mean
15
b
Right Triangle Geometric
Mean Theorem
a
b
x
h c
−
=−
; − =−
y b
x
h
b
h
y
x
c
Pythagoras
Pythagorean Theorem
a2 + b2 = c2
c
a
b
Pythagorean Inequality
Theorems
c2 < a2 + b2: acute
a
b
c > a + b : obtuse
2
2
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
c
Special Right Triangles
45°-45°-90°
ℓ = ℓ, h = ℓ √#
2
!
45°
!
2
45°
!
30°-60°-90°
h = 2s and ℓ = s √#
3
60°
2s
s
30°
s 3
Trigonometry
Ratios
opp
hyp
adj
cos A = −
hyp
opp
tan A = −
adj
sin A = −
Law of Sines
sin C
sin A
sin B
−
= −
= −
a
c
b
Law of Cosines
a2 = b2 + c2 - 2bc cos A
"
$
141
opp
#
"
c
b
$
Chapter 8
hyp
adj
a
#
Glencoe Geometry
NAME
DATE
PERIOD
Right Triangles and Trigonometry
Before the Test
Review the ideas you listed in the table at the beginning of the chapter. Cross out any
incorrect information in the first column. Then complete the table by filling in the third
column.
K
W
L
What I know…
What I want to find out…
What I learned…
See students’ work.
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 8 Study Guide and Review in the textbook.
□ I took the Chapter 8 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• While note-taking use abbreviations to use less time and room. Write neatly and place a
question mark by any information that you do not understand.
Chapter 8
142
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 8.
NAME
DATE
PERIOD
Transformations and Symmetry
Before You Read
Before you read the chapter, respond to these statements.
1. Write an A if you agree with the statement.
2. Write a D if you disagree with the statement.
Before You Read
Transformations and Symmetry
See students’ work.
• The orientation of an image that
undergoes a reflection stays the same.
• The image after a 180º rotation is equal
to the original image.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
• An image after 2 reflections in parallel
lines is equal to a translation.
• Vectors can be used to define
translations.
• Dilations can have positive or negative
scale factors.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• At the end of each lesson, write a summary of the lesson, or write in your own
words what the lesson was about.
• As you read each lesson, list examples of ways the new knowledge has been or
will be in your daily life.
Chapter 9
143
Glencoe Geometry
NAME
DATE
PERIOD
Transformations and Symmetry
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on symmetry, one fact might be the number of times a figure
maps onto itself as it rotates from 0° to 360° is called the order of symmetry. After
completing the chapter, you can use this table to review for your chapter test.
Lesson
9-1 Reflections
Fact
See students’ work.
9-2 Translations
9-3 Rotations
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
9-4 Compositions of Transformations
9-5 Symmetry
9-6 Dilations
Chapter 9
144
Glencoe Geometry
NAME
9-1
DATE
PERIOD
Reflections
What You’ll Learn
Scan the text in Lesson 9-1. Write two facts you learned
about reflections as you scanned the text.
1.
Sample
answer: To reflect a polygon in a line,
____________________________________________________
connect
the reflected vertices with line segments.
____________________________________________________
2.
Sample
answer: To reflect a point in the x-axis,
____________________________________________________
multiply
its y-coordinate by –1.
____________________________________________________
Active Vocabulary
similar polygons
Review Vocabulary Write the definition next to the term.
(Lesson 7-2)
polygons that have the same shape but not
————————————————————————————
necessarily the same size
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
————————————————————————————
New Vocabulary Label the diagram with the correct term.
line of reflection
Vocabulary Link When you look into a mirror, you see your
reflection. Explain how this relates to reflections in
geometry.
Sample answer: Each point of your image in the
————————————————————————————
mirror appears to be the same distance from each
————————————————————————————
point on the preimage.
————————————————————————————
Chapter 9
145
Glencoe Geometry
Lesson 9-1
reflect
each of the polygon’s vertices. Then
____________________________________________________
NAME
DATE
PERIOD
Lesson 9-1 (continued)
Main Idea
Draw Reflections
pp. 615–616
Details
Reflect !ABC in the line shown.
"
"'
#
#'
$
Draw Reflections in the
Coordinate Plane
pp. 616–618
$'
Describe the similarities and differences between
reflecting a point in the x-axis and reflecting a point
in the y-axis.
Similarities Sample answer: Both methods result in a
——————————————————————
point being equidistant from one of the axes on a
————————————————————————————
coordinate grid.
————————————————————————————
Differences Sample answer: To reflect a point in the
——————————————————————
x-axis, you multiply its y-coordinate by -1. To
————————————————————————————
reflect a point in the y-axis, you multiply its
————————————————————————————
x-coordinate by -1.
————————————————————————————
Helping You Remember
Sometimes it is helpful to put a geometric
concept into your own words to help you remember it. Explain in your own words how
you can reflect a point (x, y) in the line y = x.
Sample answer: To reflect a point (x, y) in the line y = x, interchange the
—————————————————————————————————————
coordinates. The reflected image is the point (y, x).
—————————————————————————————————————
Chapter 9
146
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
————————————————————————————
NAME
DATE
9-2
PERIOD
Translations
What You’ll Learn
Skim the Examples for Lesson 9-2. Predict two things you
think you will learn about translations.
1.
Sample
answer: how to draw a translation given
____________________________________________________
a____________________________________________________
figure and a translation vector
____________________________________________________
2.
Sample
answer: how to perform translations in the
____________________________________________________
coordinate
plane
____________________________________________________
Active Vocabulary
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
vector
Review Vocabulary Write the correct term next to each
definition. (Lesson 8-7)
a quantity that has both magnitude and direction
magnitude
the length of the vector from its initial point to its terminal
point
component form
a description of a vector in terms of its horizontal change x
and vertical change y from its initial point to its terminal
point
standard position
a vector that is placed in the coordinate plane with its initial
point at the origin
New Vocabulary Fill in each blank with the correct term or
phrase.
translation vector
Chapter 9
A translation maps each point to its image along a vector ,
called the translation vector, such that
segment
•
each
same length
•
this segment is also
147
joining a point and its image has the
as the vector, and
parallel
to the vector.
Glencoe Geometry
Lesson 9-2
____________________________________________________
NAME
DATE
PERIOD
Lesson 9-2 (continued)
Main Idea
Draw Reflections
pp. 615–616
Details
Draw the translation of !RST along the translation
vector.
"
v
3'
3
4'
4
5'
5
Draw Reflections in the
Coordinate Plane
pp. 616–618
Complete the following table to demonstrate how to
translate a point in the coordinate plane.
To translate a point along vector (a, b),
add a to the x-coordinate and b to the
y-coordinate.
Symbols
(x, y) → (x + a, y + b)
Example
The image of point R(2, 9) translated along
vector (1, -4) is (3, 5).
.
Helping You Remember
A good way to remember a new mathematical
term is to relate it to an everyday meaning of the same word. How is the meaning of
translation in geometry related to the idea of translation from one language to another?
Sample answer: When you translate from one language to another, you carry
—————————————————————————————————————————
over the meaning from one language to another. When you translate
—————————————————————————————————————————
a geometric figure, you carry over the figure from one position to another
—————————————————————————————————————————
without changing its basic orientation.
—————————————————————————————————————————
Chapter 9
148
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Words
NAME
9-3
DATE
PERIOD
Rotations
What You’ll Learn
Skim Lesson 9-3. Predict two things that you expect to learn
based on the headings and the Key Concept box.
1.
Sample
answer: what it means to rotate a point
____________________________________________________
about
a fixed point
____________________________________________________
____________________________________________________
2.
Sample
answer: how to draw rotations clockwise
____________________________________________________
and
counterclockwise in the coordinate plane
____________________________________________________
____________________________________________________
center of rotation
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
angle of rotation
New Vocabulary Fill in each blank with the correct term or
phrase.
A rotation about a fixed point , called the center of
rotation, through an angle of x˚ maps a point to its image
such that
•
if the point is the center of rotation, then the image and
preimage are the same point , or
•
if the point is not the center of rotation than the
image and preimage are the same distance from the
center of rotation. The measure of the angle of rotation
formed by the preimage, center of rotation, and image
points is x.
Vocabulary Link Describe some real-world examples of
rotations that you have encountered.
Sample answers: tires on a car or a bike, Ferris
————————————————————————————
wheel, CD or DVD player
————————————————————————————
————————————————————————————
————————————————————————————
Chapter 9
149
Glencoe Geometry
Lesson 9-3
Active Vocabulary
NAME
DATE
PERIOD
Lesson 9-3 (continued)
Main Idea
Draw Rotations
pp. 632–633
Details
Draw a 90˚ counterclockwise rotation of !ABC about
point D.
$'
"'
"
#
#' %
$
Draw Rotations in the
Coordinate Plane
pp. 633–634
Fill in the boxes to describe the rotation of !DEF
about the origin. Include the angle of rotation and the
direction.
180˚ clockwise
or 180˚
counterclockwise
''
&'
−8−6−4−2 0
%'
90˚ clockwise
or 270˚
counterclockwise
''
−4
−6
&' −8
''
2 4 6 8x
%
&
'
preimage
Helping You Remember
A good way to help you remember a concept in
geometry is to explain it to someone else. Suppose a classmate is having difficulty
remembering how to find the coordinates of a point after a 180˚ rotation about the
origin. How would you explain it to him?
Sample answer: Multiply the x- and y-coordinates by –1.
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 9
150
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y
8
&'
6
4
%' 2
%'
270˚ clockwise
or 90˚ counterclockwise
NAME
DATE
9-4
PERIOD
Compositions of Transformations
What You’ll Learn
Scan Lesson 9-4. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: Glide Reflections
____________________________________________________
____________________________________________________
2.
Sample
answer: Compositions of Two Reflections
____________________________________________________
____________________________________________________
center of rotation
a line such that the points of a preimage and its image are
the same distance from the line
line of reflection
the object that determines the distance and direction that a
preimage is slid to create its image
angle of rotation
the point about which a preimage is turned to create its
image
translation vector
composition of
transformations
glide reflection
Chapter 9
Review Vocabulary Match the term with its definition by
drawing a line to connect the two. (Lesson 1-5)
the amount a preimage is turned to create its image
New Vocabulary Write the correct term next to each
definition.
the result when a transformation is applied to a figure and
then another transformation is applied to its image
the composition of a translation followed by a reflection in a
line parallel to the translation vector
151
Glencoe Geometry
Lesson 9-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
NAME
DATE
PERIOD
Lesson 9-4 (continued)
Main Idea
Glide Reflections
pp. 641–642
Details
Describe the composition of transformations that lead
to !A"B"C".
$
"
$'
−8−6−4−2 0
"'
Compositions of Two
Reflections
pp. 642–644
8
6
4
# 2
−4
−6
#'−8
y
translated down 10 units, then
2 4 6 8x
#''
$''
reflected in the y-axis
"''
State the type of transformation that is achieved by
each of the following compositions.
Compositions of Transformations
Translation
Rotations
the composition
of a reflection
and a translation
the composition of
two reflections in
parallel lines
the composition
of two reflections
in intersecting
lines
Helping You Remember
Theorem 9.1 in the student text states that the
composition of two or more isometries is also an isometry. Describe what this means in
your own words, and state a property of congruence that leads to this theorem.
Sample answer: An isometry is a transformation that preserves congruence.
—————————————————————————————————————————
So, Theorem 9.1 states that the composition of two or more reflections,
—————————————————————————————————————————
translations, or rotations, results in an image that is congruent to its preimage.
—————————————————————————————————————————
This is suggested by the Transitive Property of Congruence.
—————————————————————————————————————————
Chapter 9
152
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glide Reflection
NAME
DATE
9-5
PERIOD
Symmetry
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in the lesson.
1.
Sample
answer: how to identify line and rotational
____________________________________________________
symmetries
in two-dimensional figures
____________________________________________________
2.
Sample
answer: how to identify line and rotational
____________________________________________________
symmetries
in three-dimensional figures
____________________________________________________
Active Vocabulary
line of symmetry
also called the axis of symmetry
also called the point of symmetry
order of symmetry
the number of times a figure maps onto itself as it rotates
from 0˚ to 360˚
symmetry
the characteristic of a figure if there exists a rigid motion
that maps the figure onto itself
magnitude of symmetry
the smallest angle through which a figure can be rotated so
that it maps onto itself
line symmetry
the characteristic of a figure if the figure can be mapped
onto itself by a reflection in a line
rotational symmetry
the characteristic of a figure if the figure can be mapped
onto itself by a rotation between 0˚ and 360˚
Vocabulary Link Symmetry is a word that is used in everyday
English. Find the definition of symmetry using a dictionary.
Describe how the definition of symmetry can help you
remember the mathematical definition of symmetry.
Sample answers: corresponding in shape, size, and
————————————————————————————
relative position on opposite sides of a dividing line;
————————————————————————————
A figure has symmetry when it can be mapped onto
————————————————————————————
itself, which results in a figure with the same shape
————————————————————————————
and size.
————————————————————————————
Chapter 9
153
Glencoe Geometry
Lesson 9-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
center of symmetry
New Vocabulary Write the correct term next to each
definition.
NAME
DATE
PERIOD
Lesson 9-5 (continued)
Main Idea
Symmetry in TwoDimensional Figures
pp. 653–654
Circle the figures that have rotational symmetry.
1.
2.
3.
4.
State whether the figure below has plane symmetry,
axis symmetry, both, or neither.
Step 1: Can the figure be mapped onto
itself by a reflection in a plane? yes
Step 2: Can the figure be mapped onto
itself by a rotation between 0˚ and
360˚ in a line? yes
Step 3: What kind of symmetry does
the figure have? both
Helping You Remember
What is an easy way to remember the order and
magnitude of the rotational symmetry of a regular polygon?
Sample answer: The order is the same as the number of sides. To find the
—————————————————————————————————————————
magnitude, divide 360˚ by the number of sides.
—————————————————————————————————————————
Chapter 9
154
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Symmetry in ThreeDimensional Figures
p. 655
Details
NAME
9-6
DATE
PERIOD
Dilations
What You’ll Learn
Skim the lesson. Write two things you already know about
dilations.
1.
Sample
answer: A dilation is a similarity
____________________________________________________
proportionally.
____________________________________________________
2.
Sample
answer: To find the coordinates of an
____________________________________________________
image
after a dilation centered at the origin,
____________________________________________________
multiply
the coordinates by the scale factor of the
____________________________________________________
dilation.
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
New Vocabulary State whether the dilation from figure B to
B! is an enlargement or a reduction. Then find the scale
factor of the dilation.
6
4
#
#'
enlargement
type of dilation: ________________
2.5
scale factor: ________________
Vocabulary Link Describe how the pupils of your eyes serve
as a real-world example of dilation in different lighting
conditions.
Sample answer: The pupil enlarges about the center
————————————————————————————
in low light to allow more light to enter. It reduces in
————————————————————————————
bright light to allow less light in.
————————————————————————————
Chapter 9
155
Glencoe Geometry
Lesson 9-6
transformation
that enlarges or reduces a figure
____________________________________________________
NAME
DATE
PERIOD
Lesson 9-6 (continued)
Main Idea
Draw Dilations
pp. 660–662
Details
Draw the image of !ABC under a dilation with center
D and scale factor 2.5.
"'
"
#'
#
$'
$
%
Dilations in the
Coordinate Plane
p. 662
Model a reduction by giving the coordinates of
segment R"S" and plotting it on the coordinate grid.
Tell what scale factor you used in your reduction.
Sample answer: scale: 0.5
9 y
8
7
6
5
4'
4
3
2
3
1
x
3'
−9−8−7−6−5−4−3−2−1 0
Helping You Remember
A good way to remember a new concept in
geometry is to explain it to someone else in your own words. How would you describe the
process of dilating a point (x, y) about the origin by a scale factor k?
Sample answer: Multiply the x-coordinate and y-coordinate of the point by k to
—————————————————————————————————————————
produce the image (kx, ky).
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 9
156
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
NAME
DATE
PERIOD
Transformations and Symmetry
Tie It Together
Complete the graphic organizer with a term or formula from the chapter.
y-axis
(x, y) → (-x, y)
x-axis
y=x
(x, y) → (x, -y)
(x, y) → (-y, x)
(x, y) → (x + a, y + b)
(x, y) → (kx, ky)
Reflection
Translation
Dilation
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Transformations
Composition
Glide Reflection
270°
(x, y) → (y, -x)
Chapter 9
90°
Rotation
157
(x, y) → (-y, x)
180°
(x, y) → (-x, -y)
Glencoe Geometry
NAME
DATE
PERIOD
Transformations and Symmetry
Before the Test
Now that you have read and worked through the chapter, think about what you have
learned and complete the table below. Compare your previous answers with these.
1. Write an A if you agree with the statement.
2. Write a D if you disagree with the statement.
Transformation and Symmetry
After You Read
• The orientation of an image that undergoes a reflection
stays the same.
D
• The image after a 180° rotation is equal to the original
image.
D
• An image after 2 reflections in parallel lines is equal to a
translation.
A
• Vectors can be used to define translations.
• Dilations can have positive or negative scale factors.
A
A
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 9 Study Guide and Review in the textbook.
□ I took the Chapter 9 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• On test day, look over the entire test to get an idea of its length and scope so that you
can pace yourself. Answer what you know first, skipping over material you do not know.
When finished, go back and check for errors. Don’t change an answer unless you are
certain you are correct.
Chapter 9
158
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 9.
NAME
DATE
PERIOD
Circles
Before You Read
Before you read the chapter, think about what you know about circles. List three things you
already know about them in the first column. Then list three things you would like to learn
about them in the second column.
K
W
What I know…
What I want to find out…
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
See students’ work.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• When you take notes, record real-life examples of how you can use fractions,
decimals, and percents, such as telling time and making change.
• When you take notes, listen or read for main ideas.
Then record those ideas for future reference.
Chapter 10
159
Glencoe Geometry
NAME
DATE
PERIOD
Circles
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on measuring angles and arcs, one fact might be a central angle
is an angle with a vertex in the center of the circle. After completing the chapter, you can
use this table to review for your chapter test.
Lesson
10-1 Circles and Circumference
Fact
See students’ work.
10-2 Measuring Angles and Arcs
10-3 Arcs and Chords
10-4 Inscribed Angles
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
10-5 Tangents
10-6 Secants, Tangents, and Angle
Measures
10-7 Special Segments in a Circle
10-8 Equations of a Circle
Chapter 10
160
Glencoe Geometry
NAME
DATE
10-1
PERIOD
Circles and Circumference
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in the lesson.
1.
Sample
answer: how to identify parts of a circle
____________________________________________________
2.
Sample
answer: how to find the circumference of
____________________________________________________
a____________________________________________________
circle
Active Vocabulary
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
circumference
the distance around the circle
radius
a segment with endpoints at the center and on the circle
circle
the locus or set of all points in a plane equidistant from a
given point
concentric circles
diameter
pi (π)
inscribed
two coplanar circles that have the same center
a chord which passes though the center of a circle and is
made up of collinear radii
an irrational number which by definition is the ratio of the
circumference of a circle to the diameter of the circle
descriptor give to a polygon which is drawn inside a circle
such that all of its vertices lie on the circle
center
the name used to describe the given point in the definition of
a circle
chord
a segment with endpoints on the circle
circumscribed
congruent circles
Chapter 10
New Vocabulary Write the correct term next to each
definition.
descriptor given to a circle which is drawn about a polygon
such that the circle contains all of the vertices of the polygon
two circles with congruent radii
161
Glencoe Geometry
Lesson 10-1
——————————————————————————
NAME
DATE
Lesson 10-1
PERIOD
(continued)
Main Idea
Segments in a circle
pp. 683–685
Details
Compare and contrast the pairs of special segments of
a circle in the diagram below.
A radius is not a
chord because a
radius does not
have both
endpoints on the
circle.
Chord
A diameter is a
chord that must
go through the
center of the
circle.
Diameter
Radius
The diameter of a circle is made up of two collinear
radii, thus its measure is twice that of the radius. In
addition, you can say that the radius is half of the
diameter.
The circumference of a circle is 234 inches. Determine
the radius and the diameter of the circle to the
nearest hundredth inch.
Start with
C = πd, and fill
in what you
know.
234 = πd
Solve for the
unknown.
234 = πd
d ≈ 74.48 in.
Determine the
radius.
1
r=−
(74.48)
2
r = 37.24 in.
Helping You Remember
Look up the origins of the two parts of the word
diameter in your dictionary. Explain the meaning of each part and give a term you
already know that shares the origin of that part.
Sample answer: The first part comes from dia, which means across or through,
—————————————————————————————————————————
as in diagonal. The second part comes from metron, which means measure.
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 10
162
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Circumference
pp. 685–686
NAME
10-2
DATE
PERIOD
Measuring Angles and Arcs
What You’ll Learn
Scan the text in Lesson 10-2. Write two facts you learned
about measuring angles and arcs as you scanned the text.
1.
Sample
answer: A circle has 360 degrees.
____________________________________________________
____________________________________________________
____________________________________________________
____________________________________________________
Sample
answer: An arc is a portion of a circle
____________________________________________________
defined
by two endpoints.
____________________________________________________
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
New Vocabulary Label the diagram with the terms listed at
the left.
central angle
central angle
y
arc
minor arc
#
x
minor arc
major arc
$
major arc
semicircle
"
semicircle
y
congruent arcs
#
x
congruent arcs
adjacent arcs
adjacent arcs
Chapter 10
163
Glencoe Geometry
Lesson 10-2
2.
NAME
DATE
PERIOD
Lesson 10-2 (continued)
Main Idea
Angles and Arcs
pp. 692–694
Details
Given the m∠EJD = 15, find each measure in " J in
the order specified. Justify your answer.
! = 180
mEFG
! = 15
mED
It is a semicircle
The measure of a
minor arc is equal
because EG is a
diameter.
'
to the measure of
&
%
+
! = 75
mGH
Because EHG is a
semicircle, you
(
can do 180 - 105.
its central angle.
! = 105
mEDH
)
The measure is
equal to the sum of
two adjacent arcs:
90 + 15.
! and BDC
!. The radius of " A is
Find the length of BC
5 centimeters. What is the relationship between the
two arc lengths?
#
"
%
100°
! = 8.73 cm
mBC
! = 22.69 cm
mBDC
$
Relationship: The sum of the arcs is equal to the
circumference of the circle.
Chapter 10
164
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Arc Length
p. 695
NAME
DATE
10-3
PERIOD
Arcs and Chords
What You’ll Learn
Scan Lesson 10-3. List two headings you would use to make
an outline of this lesson.
Sample
answer: Arcs and Chords
____________________________________________________
1.
——————————————————————————
Sample
answer: Bisecting Arcs and Chords
____________________________________________________
2.
——————————————————————————
Review Vocabulary In the circle provided, draw a chord which
is not a diameter using a regular line, a diameter using a
heavy line, and a radius using a dashed line. Identify 3 arcs
that were created on the circle. (Lessons 10-1 and 10-2)
Sample answers are given.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
&
Arcs
#
!
FE
!
DCB
!
DC
'
$
%
Main Idea
Arcs and Chords
pp. 701–702
Details
Find the value of x.
.
Use the definition of
congruence to write an
equation. Solve the
equation for x.
3x + 7
1
2
/
5x - 9
3x + 7 = 5x - 9
0
x=8
31
What is PO?
What is MN?
31
! and MN
!?
What realtionship exists between PO
The arcs are congruent.
————————————————————————————
Chapter 10
165
Glencoe Geometry
Lesson 10-3
Active Vocabulary
NAME
DATE
PERIOD
Lesson 10-3 (continued)
Main Idea
Bisecting Arcs
and Chords
pp. 702–704
Details
Use the diagram below to state Theorems 10.3 and
10.4, from the student book, in your own words.
"
2
/
#
1
!
!
Theorem 10.3 AB is a diameter of the circle and NP is a
! is a perpendicular bisector of NP
!.
chord, therefore, AB
Theorem 10.4 Because NO is congruent to OP, AB is
diameter of the circle.
Helping You Remember
Writing a mathematical concept in your own
words can help you better remember the concept. Describe Theorem 10.5, from the
student book, in your own words.
Sample answer: If two chords in a circle are congruent, and you draw a
—————————————————————————————————————————
segment perpendicular to the 1st chord with the center of the circle as the
—————————————————————————————————————————
other endpoint, it will be the same length as the segment perpendicular from
—————————————————————————————————————————
the 2nd chord to the center.
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 10
166
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
! is a
its perpendicular bisector, which means AB
NAME
10-4
DATE
PERIOD
Inscribed Angles
What You’ll Learn
Skim Lesson 10-4. Predict two things that you expect to
learn based on the headings and figures.
1.
Sample
answer: three ways that an angle can
____________________________________________________
be
inscribed in a circle
____________________________________________________
2.
Sample
answer: the definition of an inscribed
____________________________________________________
polygon
____________________________________________________
Active Vocabulary
Review Vocabulary Complete the chart below. (Lesson 6-1)
Regular Polygons
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3-sides
4-sides
5-sides
6-sides
triangle
quadrilateral
pentagon
hexagon
sum of interior
angles
sum of interior
angles
sum of interior
angles
sum of interior
angles
measure of
each angle
measure of
each angle
measure of
each angle
measure of
each angle
180°
60°
360°
90°
540°
108°
720°
120°
inscribed angle
an angle which has a vertex on a circle and sides that
————————————————————————————
contain chords on the circle
————————————————————————————
intercepted arc
an arc which has endpoints on the sides of an inscribed
————————————————————————————
angle and lies in the interior of an inscribed angle
————————————————————————————
Chapter 10
167
Glencoe Geometry
Lesson 10-4
New Vocabulary Write the definition next to each term.
NAME
DATE
PERIOD
Lesson 10-4 (continued)
Main Idea
Inscribed Angles
pp. 709–711
Details
Find the indicated measure for each figure.
"
1.
2.
3
4
22°
#
$
2
124°
m∠BAC =
Angles of Inscribed
Polygons
pp. 711–712
62
!=
mRS
44
Provide details for Theorem 10.9 in the student book
by defining each boldface word.
a polygon with 4
sides
a polygon which is
drawn inside a circle
such that all of its
vertices lie on the circle
two angles whose
sum is 180°
any two angles in a
quadrilateral that are
not consecutive
Helping You Remember
Describe how you could make a sketch that
would help you remember the relationship between the measure of an inscribed angle
and the measure of its intercepted arc.
Sample answer: Draw a diameter of a circle to divide it into two semicircles.
—————————————————————————————————————————
Inscribe an angle in one of the semicircles; this angle will intercept the other
—————————————————————————————————————————
semicircle. From your sketch, you can see that the inscribed angle is a right
—————————————————————————————————————————
angle. The measure of the semicircle arc is 180°, so the measure of the
—————————————————————————————————————————
inscribed angle is half the measure of its intercepted arc.
—————————————————————————————————————————
Chapter 10
168
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
If a quadrilateral is inscribed
in a circle, then its opposite angles
are supplementary.
NAME
10-5
DATE
PERIOD
Tangents
What You’ll Learn
Skim the Examples for Lesson 10-5. Predict two things you
think you will learn about tangents.
1.
Sample
answer: how to identify common tangents
____________________________________________________
____________________________________________________
2.
Sample
answer: how to use tangents to find
____________________________________________________
measures
____________________________________________________
Active Vocabulary
Review Vocabulary Use the Pythagorean Theorem to
determine if the triangles with the specified sides are right
triangles. (Lesson 8-2)
a = 6, b = 8, c = 10 a = 4, b = 5, c = 41 a = 9, b = 40, c = 41
yes
no
yes
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
New Vocabulary Fill in each blank with the correct term or
phrase.
tangent
line
a
in the same plane as a
circle
exactly
point of tangency
that
one
the point where a
intersects
the circle in
point
tangent
line intersects a
circle
Vocabulary Link Use a dictionary to look up the origin of the
word tangent. How can the origin of this word help you
remember the definition of a tangent line?
Sample answer: The word tangent comes from the
Latin word tangere, which means “to touch.” A line
————————————————————————————
that is tangent to a circle just “touches” the circle in
————————————————————————————
one point.
————————————————————————————
Chapter 10
169
Glencoe Geometry
Lesson 10-5
————————————————————————————
NAME
DATE
PERIOD
Lesson 10-5 (continued)
Main Idea
Tangents
pp. 718–720
Details
Find the value of x. Assume that segments that
appear to be tangent are tangent. Show your work in
each box as indicated.
Identify the length of each side
of the right triangle.
"
x
x
Leg 1 =
8
12
Hyp =
x
Leg 2 =
12
x+8
Use the Pythagorean Theorem to write an equation, then
solve for x.
Circumscribed
Polygons
p. 721
Fill in the missing measurements in the diagram
below in the order specified. Justify each
measurement.
1. The measure of
3. The measure of
this segment is 12
because of
Theorem 10.11.
2. The measure of
this segment is 4
because of
Theorem 10.11.
Chapter 10
170
this segment is 7
because of
segment addition.
12
4
11
4. The measure of
this segment is 7
because of
Theorem 10.11.
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x2 + 122 = (x + 8)2
x2 + 144 = x2 + 16x + 64
80 = 16x
x=5
NAME
DATE
10-6
PERIOD
Secants, Tangents, and Angle Measures
What You’ll Learn
Skim the lesson. Write two things you already know about
secants, tangents, and angle measures.
1.
Sample
answer: The measure of an inscribed
____________________________________________________
2.
Sample
answer: A tangent line intersects a circle
____________________________________________________
at
one point.
____________________________________________________
Active Vocabulary
Review Vocabulary Determine the measure of each angle in
the diagram. Justify your answers. (Lesson 1-5)
1
2
3
125°
Angle 1 is 55° because
it is supplementary to
the 125° angle.
Angle 2 is 125° because
it is a vertical angle
with the given angle.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Angle 3 is 55° because
it is a vertical angle
with angle 1.
New Vocabulary Write the definition next to the term.
secant
a line which intersects a circle in exactly two points
————————————————————————————
————————————————————————————
Main Idea
Intersections On or
Inside a Circle
pp. 727–729
Details
Find the value of x. Show your work in each box as
indicated.
&
"
136°
Determine the measure of ∠EAF.
)
x°
1 ( 62
+ 136 ) = 99
m∠EAF = −
62°
$
Determine the measure of ∠CAF.
81
'
Chapter 10
2
171
Glencoe Geometry
Lesson 10-6
angle
is half the measure of its intercepted arc.
____________________________________________________
NAME
DATE
PERIOD
Lesson 10-6 (continued)
Main Idea
Intersections Outside
a Circle
Label each diagram and explain how you would
determine the measure of ∠1 in each diagram.
pp. 729–731
tangent
intercepted arc
1
secant
The measure of ∠1 is one half the measure
of the intercepted arc.
intercepted arc
secant
secant
The measure of ∠1 is one half the measure
of the difference of the intercepted arc.
Helping You Remember
Some students have trouble remembering the
difference between a secant and a tangent. What is an easy way to remember which is
which?
Sample answer: A secant cuts a circle, while a tangent just touches it at one
—————————————————————————————————————————
point. You can associate tangent with touches because they both start with t.
—————————————————————————————————————————
Then associate secant with cuts.
—————————————————————————————————————————
Chapter 10
172
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
intercepted arc
NAME
10-7
DATE
PERIOD
Special Segments in a Circle
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in the lesson.
1.
Sample
answer: how to find the measure of
____________________________________________________
segments
that intersect inside a circle
____________________________________________________
____________________________________________________
2.
Sample
answer: how to find the measure of
____________________________________________________
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
Review Vocabulary Write the definition next to each term.
(Lessons 10-1, 10-5, and 10-6)
chord
a segment with endpoints on the circle
secant
a line which intersects a circle in exactly two points
tangent
————————————————————————————
————————————————————————————
the point where a line intersects a circle
————————————————————————————
New Vocabulary Match each term with its definition by
drawing a line to connect the two.
Chapter 10
tangent segment
a segment formed when two chords intersect inside a circle
secant segment
a segment of a secant line that has exactly one endpoint on
the circle
external secant
segment
a segment of a tangent line that has exactly one endpoint on
the circle
chord segment
a segment of a secant line that has an endpoint which lies in
the exterior of the circle
173
Glencoe Geometry
Lesson 10-7
segments
that intersect outside a circle
____________________________________________________
NAME
DATE
PERIOD
Lesson 10-7 (continued)
Main Idea
Segments Intersecting
Inside a Circle
pp. 736–737
Details
Use Theorem 10.15 from the student book to find x.
RP = 12, PS = 3x + 5, QP = 15, PT = 24
3
2
1
5
4
RP ! PS = QP ! PT
12
!
3x + 5
=
15
36x + 60 =
360
Segments Intersecting
Outside a Circle
pp. 738–739
Chapter 10
=
1
8−
3
Compare and contrast Theorem 10.16 with
Theorem 10.15 from the student book.
How are they the same?
How are they different?
Both define a
relationship between the
segments created when
secants/chords intersect.
Both relationships
equate the products of
segments.
10.16 equates the
products of the two
pieces of the divided
chords. 10.16 equates
the products of the
whole segment with the
part of the segment
outside the circle.
174
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x
24
!
NAME
10-8
DATE
PERIOD
Equations of Circles
What You’ll Learn
Skim the lesson. Write two things you already know about
equations of circles.
1.
Sample
answer: The Pythagorean Theorem is
____________________________________________________
2
a____________________________________________________
+ b2 = c2.
2.
Sample
answer: The Distance Formula is
____________________________________________________
"""""""""
(x1 - x2)2 + (y1 - y2)2 .
√
____________________________________________________
Active Vocabulary
Review Vocabulary Use the Distance Formula to find the
distance between the given pairs of points on the coordinate
plane. (Lesson 1-5)
(1, 2) and (9, 11)
√""
145
(1, 2) and (-7, -7) (9, 11) and (-7, -7)
√""
145
2 √""
145
compound locus
a locus that satisfies more than one set of conditions
————————————————————————————
————————————————————————————
Vocabulary Link Previously, you wrote the equation of a line
when given the graph of the line. Compare and contrast this
process with writing the equation of a circle given the graph
of the circle.
Sample answer: Both a line and a circle have a
————————————————————————————
general form. When writing equations of each, you
————————————————————————————
must identify two important characteristics of the
————————————————————————————
graph. In a line, you need the y-intercept and the
————————————————————————————
slope. In a circle, you need the center and the radius.
————————————————————————————
Chapter 10
175
Glencoe Geometry
Lesson 10-8
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
New Vocabulary Fill in the blank with the correct term or
phrase.
NAME
DATE
PERIOD
Lesson 10-8 (continued)
Main Idea
Equation of a Circle
pp. 744–745
Details
Write the equation of a circle that has a center at
(1, 3) and passes though (6, -3).
Find the length of the radius using the center and the
point of the circle.
""
(6 - 1) 2 + (-3 - 3) 2 = √"""""
5 2 + (-6) 2 = √61
√""""""""
Determine h, k, and r2.
h=
1
k=
3
r2 =
61
)2 =
61
Write the equation of the circle.
(x Graph Circles
pp. 745–746
1
)2 + (y -
3
Graph the circle given by the equation
(x + 4)2 + (y - 2)2 = 16.
(x - (-4))2 + (y - 2)2 = 16
Identify the center. (-4, 2)
Identify the radius.
y
4
Use the center and radius
to identify four points on
the circle.
(-8, 2), (0, 2),
0
x
(-4, 6), (-4, -2)
Chapter 10
176
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Rewrite the equation in standard form.
NAME
DATE
PERIOD
Circles
Tie It Together
Name each part of the circle shown below.
4FDBOU
$PODFOUSJD$JSDMFT
"
#
!
4FNJDJSDMF #&
0
'
$IPSE
$FOUSBM
"OHMF
*OTDSJCFE
"OHMF
$
3BEJVT
%
%JBNFUFS
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
(
1PJOUPG5BOHFODZ
&
!
&$
!
#&$
.JOPS"SD
.BKPS"SD
5BOHFOU-JOF
Chapter 10
177
Glencoe Geometry
NAME
DATE
PERIOD
Circles
Before the Test
Review the ideas you listed in the table at the beginning of the chapter. Cross out any
incorrect information in the first column. Then complete the table by filling in the third
column.
K
W
L
What I know…
What I want to find out…
What I learned…
See students’ work.
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 10 Study Guide and Review in the textbook.
□ I took the Chapter 10 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• To prepare to take lecture notes, make a column to the left about 2 inches wide. Use
this column to write additional information from your text, place question marks, and to
summarize information.
Chapter 10
178
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 10.
NAME
DATE
PERIOD
Areas of Parallelograms and Triangles
Before You Read
Before you read the chapter, think about what you know about areas of parallelograms and
triangles. List three things you already know about them in the first column. Then list
three things you would like to learn about them in the second column.
K
W
What I know…
What I want to find out…
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
See students’ work.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• Write down questions that you have about what you are reading in the lesson.
Then record the answer to each question as you study the lesson.
• A visual (graph, diagram, picture, chart) can present information in a concise,
easy-to-study format.
Clearly label your visuals and write captions when needed.
Chapter 11
179
Glencoe Geometry
NAME
DATE
PERIOD
Areas of Parallelograms and Triangles
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on areas of circles and sectors, one fact might be a sector of a
circle is a region of a circle bounded by a central angle and its intercepted arc. After
completing the chapter, you can use this table to review for your chapter test.
Lesson
11-1 Areas of Parallelograms and
Triangles
Fact
See students’ work.
11-2 Areas of Trapezoids, Rhombi, and
Kites
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
11-3 Areas of Circles and Sectors
11-4 Areas of Regular Polygons and
Composite Figures
11-5 Areas of Similar Figures
Chapter 11
180
Glencoe Geometry
NAME
11-1
DATE
PERIOD
Areas of Parallelograms and Triangles
What You’ll Learn
Skim Lesson 11-1. Predict two things you expect to learn
based on the headings and the Key Concept box.
1.
Sample
answer: how to use the formula for finding
____________________________________________________
2.
Sample
answer: how to use the formula for finding
____________________________________________________
the
area of a triangle
____________________________________________________
Active Vocabulary
Review Vocabulary Define parallelogram in your own words.
(Lesson 6-2)
Sample answer: a quadrilateral with both pairs of
————————————————————————————
opposite sides parallel
————————————————————————————
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
New Vocabulary Match each term with its definition.
base of a parallelogram
height of a parallelogram
base of a triangle
height of a triangle
the perpendicular distance between any two parallel bases
any side of a triangle
any side of a parallelogram
the length of an altitude drawn to a given base
Vocabulary Link Formula is a word that is used in everyday
English. This lesson introduces two new formulas. Find the
definition of formula using a dictionary. Explain how its
English definition can help you understand the meaning of
formula in mathematics.
Sample answer: A formula is a fixed procedure for
————————————————————————————
doing something. In mathematics, it is a rule for
————————————————————————————
finding a value (usually expressed with algebraic
————————————————————————————
symbols).
————————————————————————————
Chapter 11
181
Glencoe Geometry
Lesson 11-1
the
area of a parallelogram
____________________________________________________
NAME
DATE
PERIOD
Lesson 11-1 (continued)
Main Idea
Areas of Parallelograms
pp. 763–764
Details
Find the perimeter and area of the parallelogram
below. Round to the nearest tenth if necessary.
22 cm
10 cm
6 cm
The perimeter measures
64
cm.
8
The height of the parallelogram measures
The area of the parallelogram measures
Areas of Triangles
pp. 765–766
176
cm.
cm2.
Use the organizer below to determine how the area of
a triangle is related to the area of a rectangle.
Area of a
rectangle?
A = bh
Now you have
two
‘s.
Conclusion: The rectangle cut in half
How do these
triangles
relate to the
rectangle?
gives two triangles. Therefore, the
area of a triangle is equal to one half
the area of a rectangle.
1
Area of Triangle = −
bh
2
Helping You Remember
A good way to remember a new formula in
geometry is to relate it to a formula you already know. How can you use the formula for
the area of a rectangle to help you remember the formula for the area of a
parallelogram?
Sample answer: Due to the Area Addition Postulate, a parallelogram can be
—————————————————————————————————————————
made into a rectangle, making their area formulas exactly the same.
—————————————————————————————————————————
Chapter 11
182
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Draw a
diagonal.
NAME
11-2
DATE
PERIOD
Areas of Trapezoids, Rhombi, and Kites
What You’ll Learn
Scan the text in Lesson 11-2. Write two facts you learned
about trapezoids, rhombi, and kites as you scanned the text.
1.
Sample
answer: the definition of a trapezoid
____________________________________________________
____________________________________________________
2.
Sample
answer: how to use the formula for finding
____________________________________________________
the
area of a trapezoid
____________________________________________________
parallelogram
trapezoid
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
base of a triangle
Review Vocabulary Fill in each blank with the correct term or
phrase. (Lessons 6-2, 6-6, and 11-1)
The base of a triangle
is any side of a triangle.
A quadrilateral with both pairs of opposites sides is known
as a(n) parallelogram .
A(n) trapezoid
of parallel sides.
is a quadrilateral with exactly one pair
New Vocabulary Define height of a trapezoid in your own
words.
Sample answer: The height of a trapezoid is the
————————————————————————————
perpendicular distance between its two bases.
————————————————————————————
Vocabulary Link Area is a word that is used in everyday
English. Find the definition of area using a dictionary.
Explain how its English definition can help you understand
its meaning in mathematics.
Sample answer: Area is any particular extent of space
————————————————————————————
or surface. In mathematics, it is a quantitative
————————————————————————————
measure of a plane or curved surface.
————————————————————————————
Chapter 11
183
Glencoe Geometry
Lesson 11-2
Active Vocabulary
NAME
DATE
PERIOD
Lesson 11-2 (continued)
Main Idea
Areas of Trapezoids
pp. 773–774
Details
Derive the formula for the area of a trapezoid.
b1
Take a basic
trapezoid and make a
glide reflection of it.
h
b2
h
b2
These two congruent trapezoids
together make one parallelogram.
b1
Area of parallelogram =
(b1 + b2)h
Therefore, the formula for
the area of a trapezoid
1
= −(b1 + b2)h.
2
In the formula for finding the area of a rhombus or
1
kite, A = −
d1d2, what do d1 and d2 represent?
2
the lengths of the diagaonals
————————————————————————————
Find the area of the kite.
10 in.
18 in.
2
The area of the kite is 90 in .
Helping You Remember
A good way to remember a new geometric
formula is to state it in words. Write a short sentence that tells how to find the area of a
trapezoid in a way that is easy to remember.
Sample answer: To find the area of a trapezoid, add the bases, multiply their
—————————————————————————————————————————
sum by its height, then divide that product in half.
—————————————————————————————————————————
Chapter 11
184
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Areas of Rhombi
and Kites
pp. 775–776
NAME
DATE
11-3
PERIOD
Areas of Circles and Sectors
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about areas of circles and sectors.
1.
Sample
answer: how to find the area of circles
____________________________________________________
____________________________________________________
2.
Sample
answer: how to find the area of sectors of
____________________________________________________
circles
____________________________________________________
base of a triangle
central angle
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
height of a triangle
height of a trapezoid
arc
base of a parallelogram
Review Vocabulary Write the correct term next to each
definition. (Lessons 10-2, 11-1, and 11-2)
any side of a triangle
an angle with a vertex in the center of a circle and with
sides that contain two radii of the circle
the length of an altitude drawn to a given base of a triangle
the distance between the two bases of a trapezoid
a portion of a circle defined by two endpoints
any side of a parallelogram
New Vocabulary Define the sector of a circle in your own
words.
Sample answer: A sector of a circle is a region of
————————————————————————————
a circle bounded by a central angle and its
————————————————————————————
intercepted arc.
————————————————————————————
Chapter 11
185
Glencoe Geometry
Lesson 11-3
Active Vocabulary
NAME
DATE
PERIOD
Lesson 11-3 (continued)
Main Idea
Areas of Circles
pp. 782–783
Details
Find the area of each circle. Round to the nearest
tenth.
1.
2.
7 in.
18 cm
254.3 cm2
Areas of Sectors
pp. 783–784
153.9 in2
Use the organizer below and fill in the parts of the
proportion that can be used to find the area of a
sector.
The ratio of the area A of a sector to the area of
the whole circle, πr2, is equal to the ratio of the
degree measure of the intercepted arc x to 360.
A
=
x
Use the completed proportion to find the area of the shaded
sector. Round to the nearest tenth.
A ≈ 7.7 square units
55°
"
4 units
Helping You Remember
A good way to remember something is to explain
it to someone else. Suppose your classmate Joelle is having trouble remembering which
formula is for circumference and which is for area. How can you help her?
Sample answer: Circumference is measured in linear units, while area is
—————————————————————————————————————————
measured in square units, so the formula containing r2 must be one for area.
—————————————————————————————————————————
Chapter 11
186
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
360
πr2
NAME
11-4
DATE
PERIOD
Areas of Regular Polygons and Composite
Figures
What You’ll Learn
Skim the Examples for Lesson 11-4. Predict two things you
think you will learn about areas of regular polygons and
composite figures.
1.
Sample
answer: how to identify segments and
____________________________________________________
angles
in regular polygons
____________________________________________________
2.
Sample
answer: how to use the formula for the
____________________________________________________
area
of a regular polygon
____________________________________________________
Active Vocabulary
Review Vocabulary Answer the questions about the elements
of the diagram below. (Lesson 10-2)
#
"
$
central angle
2. The portion of the circle that is defined by endpoints B
arc
and C is known as a(n)
.
New Vocabulary Match each term with its definition.
apothem
the radius of the circle that is circumscribed about the
regular polygon
radius of a regular polygon
the center of the circle in which the regular polygon is
inscribed
center of a regular polygon
has its vertex at the center of the polygon and its sides pass
through consecutive vertices of the polygon
central angle of a polygon
Chapter 11
a segment drawn perpendicular to a side of a regular
polygon
187
Glencoe Geometry
Lesson 11-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. Angle A is what type of angle?
NAME
DATE
PERIOD
Lesson 11-4 (continued)
Main Idea
Areas of Regular
Polygons
pp. 791–793
Details
Identify the parts of the regular polygon.
3
8
2
4
7
6
5
Name a central
angle. Sample
What is the center?
V
answer: ∠QVR
Name an apothem.
Name a radius.
Sample answer:
−−
VR
Areas of Composite
Figures
pp. 793–794
Sample answer:
−−
VW
1
aP, where A is the area of a
Using the formula A = −
2
regular polygon, a is the apothem, and P is the
perimeter, find the area of the regular octagon below.
9 in.
2
The area of the regular octagon is 396 in .
Helping You Remember
Rolando is having trouble remembering when to
subtract an area when finding the area of a composite figure. How can you help him
remember?
Sample answer: Subtraction is usually needed when part of the figure is
—————————————————————————————————————————
shaded and part is not shaded.
—————————————————————————————————————————
Chapter 11
188
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
11 in.
NAME
DATE
11-5
PERIOD
Areas of Similar Figures
What You’ll Learn
Scan Lesson 11-5. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: Areas of Similar Figures
____________________________________________________
____________________________________________________
2.
Sample
answer: Scale Factors in Similar Figures
____________________________________________________
____________________________________________________
Active Vocabulary
Review Vocabulary Define similar polygons in your own
words. (Lesson 7-2)
Sample answer: Two polygons are similar if all their
————————————————————————————
corresponding angles are congruent, and all of their
————————————————————————————
————————————————————————————
Fill in each blank with the correct term or phrase.
(Lessons 10-1 and 11-3)
segment of a circle
sector of a circle
inscribed
A slice of pie would be an example of a(n) sector of a circle.
A figure or shape enclosed by another geometric shape or
inscribed
figure is
.
A(n) segment of a circle is the part of the interior of a
circle bound by a chord and an arc.
Vocabulary Link Similar is a word that is used in everyday
English. Find the definition of similar using a dictionary.
Explain how its English definition can help you understand
its meaning in mathematics.
Sample answer: Similar means to have a likeness or
————————————————————————————
a resemblance. In mathematics, it has to do with the
————————————————————————————
specific likeness of figures or polygons.
————————————————————————————
Chapter 11
189
Glencoe Geometry
Lesson 11-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
corresponding sides are proportional.
NAME
DATE
PERIOD
Lesson 11-5 (continued)
Main Idea
Areas of Similar Figures
pp. 802–803
Details
For the pair of similar triangles below, find the area
of !DEF.
D
9 yd
A
C
Scale Factors and
Missing Measures in
Similar Figures
pp. 803–804
F
6 yd
E
A = 6.75 yd2
B
A = 3 yd2
Complete the missing parts of the proportion below.
If you know the areas of two similar geometric shapes, you
can use them to find the scale factor between them.
9
#
Area of
:
$
ABCD = 20 cm2
"
8
%
;
Area of
WXYZ = 45 cm2
area of
ABCD
−−
= k2
area of
20
−
45
= k2
4
−= k
9
2
WXYZ
The scale factor,
k=
2
−
3
.
Helping You Remember
Anthony thinks that similar figures are equal
figures. How can you explain to him that similar and equal are not the same?
Sample answer: Tell him that similar figures will have the same shape, but they
—————————————————————————————————————————
will not be the same size. Equal figures, otherwise known as congruent figures,
—————————————————————————————————————————
have both the same shape and the same size.
—————————————————————————————————————————
Chapter 11
190
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Use k as the scale factor.
NAME
DATE
PERIOD
Areas of Polygons and Circles
Tie It Together
Complete the graphic organizer with a formula from the chapter and identify the variables
in the diagram.
Figure
Parallelogram
Formula
Diagram
A = bh
h
b
Triangle
1
A=−
bh
2
h
b
Trapezoid
1
A=−
h(b1 + b2)
b1
2
h
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b2
Rhombus
Kite
1
A=−
d1d2
Circle
A = πr2
2
d1
d2
r
Sector
x
A=−
πr2
360
r
x°
Regular Polygon
1
A= −
aP
2
1
a
Chapter 11
191
Glencoe Geometry
NAME
DATE
PERIOD
Areas of Polygons and Circles
Before the Test
Review the ideas you listed in the table at the beginning of the chapter. Cross out any
incorrect information in the first column. Then complete the table by filling in the third
column.
K
W
L
What I know…
What I want to find out…
What I learned…
See students’ work.
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 11 Study Guide and Review in the textbook.
□ I took the Chapter 11 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• If possible, rewrite your notes. Not only can you make them clearer and neater,
rewriting them will help you remember the information.
Chapter 11
192
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 11.
NAME
DATE
PERIOD
Extending Surface Area and Volume
Before You Read
Before you read the chapter, think about what you know about surface area and volume.
List three things you already know about extending surface area and volume in the first
column. Then list three things you would like to learn about in the second column.
K
W
What I know…
What I want to find out…
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
See students’ work.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• When you take notes, listen or read for main ideas.
Then record those ideas for future reference.
• To help you organize data, create study cards when taking notes, recording
and defining vocabulary words, and explaining concepts.
Chapter 12
193
Glencoe Geometry
NAME
DATE
PERIOD
Extending Surface Area and Volume
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on volumes of pyramids and cones, one fact might be pyramids
1
use −
of the area of base in the formula for volume. After completing the chapter, you can
3
use this table to review for your chapter test.
Lesson
Fact
12-1 Representations of Three-Dimensional See students’ work.
Figures
12-2 Surface Areas of Prisms and
Cylinders
12-3 Surface Areas of Pyramids and Cones
12-4 Volumes of Prisms and Cylinders
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
12-5 Volumes of Pyramids and Cones
12-6 Surface Areas and Volumes of
Spheres
12-7 Spherical Geometry
12-8 Congruent and Similar Solids
Chapter 12
194
Glencoe Geometry
NAME
PERIOD
Representations of Three-Dimensional Figures
What You’ll Learn
Skim the Examples for Lesson 12-1. Predict two things you
think you will learn about representations of threedimensional figures.
1.
Sample
answer: how to sketch a solid when given
____________________________________________________
its
dimensions
____________________________________________________
2.
Sample
answer: how to identify cross sections of
____________________________________________________
various
solids
____________________________________________________
Active Vocabulary
Review Vocabulary Define arc of a circle in your own words.
(Lesson 11-3)
Sample answer: a portion of a circle defined by two
————————————————————————————
endpoints
————————————————————————————
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
cross section
isometric view
New Vocabulary Write the correct term next to each
definition.
the intersection of a solid and a plane
a corner view of a three-dimensional geometric solid on twodimensional paper
Vocabulary Link Cross section is an expression that is used in
everyday English. Either find the definition of cross section
using a dictionary or think of how you have heard it used
outside of geometry. Explain any similarities between its
everyday use and its use in mathematics.
Sample answer: Cross section is often used to refer
————————————————————————————
to a sample section or selection of a certain
————————————————————————————
population during a study of characteristics or
————————————————————————————
opinions of people. In mathematics, it refers to a
————————————————————————————
specific view or section when studying the
————————————————————————————
characteristics of solid figures.
————————————————————————————
Chapter 12
195
Glencoe Geometry
Lesson 12-1
12-1
DATE
NAME
DATE
PERIOD
Lesson 12-1 (continued)
Main Idea
Draw Isometric Views
pp. 823–824
Details
Complete the graphic organizer below.
two different ways to draw an isometric sketch
of a solid.
Draw an isometric view
from the dimensions.
Investigate Cross
Sections
p. 824
Draw an isometric view
from an orthographic
drawing.
Name the different cross sections of a cylinder. State
how the plane would have to intersect the cylinder to
get that particular cross section.
1.
circle;
the plane intersects the cylinder
____________________________________________
parallel
to the circular base
____________________________________________
ellipse;
the plane intersects the cylinder
____________________________________________
at
an angle
____________________________________________
3.
rectangle;
the plane intersects the cylinder
____________________________________________
perpendicular
to the circular base
____________________________________________
Helping You Remember
Look up the word isometry in the dictionary.
Compare its definition with the definition of corner view and perspective view. Why are
corner views considered isometric views, but three-dimensional views not considered
isometric views?
Sample answer: An isometry is a distance preserving transformation. Corner
—————————————————————————————————————————
views represent the actual distances of figures, while three-point perspective
—————————————————————————————————————————
views represent the apparent distances of figures.
—————————————————————————————————————————
Chapter 12
196
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2.
NAME
12-2
DATE
PERIOD
Surface Areas of Prisms and Cylinders
What You’ll Learn
Scan Lesson 12-2. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: Lateral Areas and Surface Areas
____________________________________________________
of
Prisms
____________________________________________________
2.
Sample
answer: Lateral Areas and Surface Areas
____________________________________________________
Active Vocabulary
Lesson 12-2
of
Cylinders
____________________________________________________
New Vocabulary Label the elements of this oblique
pentagonal prism with the correct terms.
$
"
#
%
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
&
altitude
1. Arrow “A” is pointing to a(n)
lateral edge
bases
.
lateral edge
2. Arrow “B” is pointing to
lateral face
3. Arrow “C” is pointing to a(n)
base edge
.
base edge
4. Arrow “D” is pointing to a(n)
altitude
.
bases
5. Arrow “E” is pointing to a(n)
lateral face
.
.
Fill in each blank with the correct term or phrase
lateral area
axis
Chapter 12
axis
A(n)
of a cylinder is the segment with
endpoints that are centers of the circular bases.
The lateral area of a prism is the sum of the areas of the
lateral faces.
197
Glencoe Geometry
NAME
DATE
PERIOD
Lesson 12-2 (continued)
Main Idea
Lateral Areas and
Surface Areas of Prisms
pp. 830–831
Details
Complete the diagram to show the parts of the area
formulas for prisms.
P = perimeter of
the base
h = height of
the prism
The lateral area
L is L = Ph.
B = area of
the base
The surface area
S is S = L + 2B.
Given the base edges of the prism above are: 8, 6, 5, and 5
inches, and that the height of the prism is 10 inches, find its
lateral area. 240 in2
The cylindrical canister below has a surface area of
33 in2. Use the formula S = 2πrh + 2πr2.
1.5 in.
What is the approximate height of the
canister? 2 in.
h
Helping You Remember
A good way to remember a new mathematical
term is to relate it to an everyday use of the same word. How can the way the word
lateral is used in sports help you remember the meaning of the lateral area of a solid?
Sample answer: In football, a pass thrown to the side is called a lateral pass. In
—————————————————————————————————————————
geometry, the lateral area is the “side area,” or the sum of the areas of all the
—————————————————————————————————————————
side surfaces of the solid (the surfaces other than the bases).
—————————————————————————————————————————
Chapter 12
198
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lateral Areas and
Surface Areas of
Cylinders
pp. 832–833
NAME
12-3
DATE
PERIOD
Surface Areas of Pyramids and Cones
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about surface areas of pyramids and cones.
1.
Sample
answer: how to find lateral areas and
____________________________________________________
surface
areas of pyramids
____________________________________________________
____________________________________________________
2.
Sample
answer: how to find lateral areas and
____________________________________________________
surface
areas of cones
____________________________________________________
____________________________________________________
Review Vocabulary Label the following solids as right or
oblique. (Lesson 12-1)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1.
2.
1.
right
2.
right
3.
oblique
3.
Lesson 12-3
Active Vocabulary
New Vocabulary Match each term with its definition.
regular pyramid
the height of each lateral face of a regular pyramid
slant height
a figure where the axis of a cone is also its altitude
right cone
oblique cone
Chapter 12
a figure with a base that is a regular polygon and the
altitude has an endpoint at the center of the base
a figure where the axis of a cone is not its altitude
199
Glencoe Geometry
NAME
DATE
PERIOD
Lesson 12-3 (continued)
Main Idea
Lateral Area and
Surface Area of
Pyramids
pp. 838–840
Details
Use the formulas below to find the lateral and the
surface area of the regular pyramid pictured.
1
Lateral area, L = −
P", where P is the
2
perimeter of the base and " is the
slant height.
99 cm2
L=
1
Surface area, S = −
P" + B, where
2
P is the perimeter of the base,
" is the slant height, and
B is the area of the base.
121.4 cm2
S=
Lateral Area and
Surface Area of Cones
pp. 840–842
! = 9 cm
2.8 cm
3 cm
8 cm
Follow the steps to find the surface area of a cone.
cone height
h = 21 cm
surface area
2
S = πr! + πr
1. Use the Pythagorean Theorem to find the slant height of
the cone. Round your answer to the nearest tenth.
21.2 cm
2. Use the formula above to find the cone’s surface area.
Round to the nearest centimeter. 228 cm2
Helping You Remember
One way to remember a new formula is to relate
it to a formula you already know. Explain how the formulas for the lateral areas of a
pyramid and cone are similar.
Sample answer: Both formulas say that the lateral area includes the distance
—————————————————————————————————————————
around the base multiplied by the slant height.
—————————————————————————————————————————
Chapter 12
200
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
radius
r = 3 cm
slant height
"=?
NAME
12-4
DATE
PERIOD
Volumes of Prisms and Cylinders
What You’ll Learn
Skim Lesson 12-4. Predict two things that you expect to
learn based on the headings and the Key Concept box.
1.
Sample
answer: how to use the formula for finding
____________________________________________________
the
volume of a prism
____________________________________________________
____________________________________________________
2.
Sample
answer: how to apply Cavalieri’s Principle
____________________________________________________
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
isometric
circle
axis
lateral area
oblique
Chapter 12
Review Vocabulary Fill in each blank with the correct term or
phrase. (Lessons 12-1 through 12-3)
What shape describes the cross section
circle
of this sphere?
axis
A(n)
of a cylinder is the segment with
endpoints that are centers of the circular bases.
When the axis of a cone is not its altitude this represents
oblique
a(n)
cone.
lateral area
The
the lateral faces.
of a prism is the sum of the areas of
This type of paper is used to draw a(n)
view of a solid.
201
isometric
Glencoe Geometry
Lesson 12-4
Active Vocabulary
NAME
DATE
PERIOD
Lesson 12-4 (continued)
Main Idea
Volume of Prisms
p. 847
Details
Complete the steps to finding the volume of the prism
below.
Use the Pythagorean Theorem to find
the area of the triangular base, B. Round
to the nearest cubic inch. B = 50 in2
10 in.
10 in.
16 in.
To find the volume, V of the prism,
multiply the base, B by the height, h
of the prism.
V=
Volume of Cylinders
pp. 848–849
14.2 in.
800 in3
Use the formula V = πr2h to find the volume of the
10 cm
cylinder shown.
4 cm
Helping You Remember
A good way to remember a mathematical concept
is to explain it to someone else. Suppose that your younger sister, who is in eighth
grade, is having trouble understanding why square units are used to measure area, but
cubic units are needed to measure volume. How can you explain this to her in a way
that will make it easy for her to understand and remember the correct units to use?
Sample answer: Area measures the amount of space inside a two-dimensional
—————————————————————————————————————————
figure, while volume measures the amount of space inside a three-dimensional
—————————————————————————————————————————
figure. A two-dimensional figure can be covered with small squares, which
—————————————————————————————————————————
represent square units. A three-dimensional figure can be filled with small
—————————————————————————————————————————
cubes, which represent cubic units.
—————————————————————————————————————————
Chapter 12
202
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
V = 502.4 cm3
NAME
DATE
12-5
PERIOD
Volumes of Pyramids and Cones
What You’ll Learn
Skim the lesson. Write two things you already know about
volumes of pyramids and cones.
1.
Sample
answer: There are different types of
____________________________________________________
pyramids
based upon the shape of the base.
____________________________________________________
____________________________________________________
2.
Sample
answer: Some cones are straight up and
____________________________________________________
down
(right cones), and some are slanted in one
____________________________________________________
direction
or another (oblique cones).
____________________________________________________
Review Vocabulary Fill in each blank with the correct term or
phrase. (Lessons 11-1 through 11-5)
A.
B.
C.
D.
area of a trapezoid
Polygon A represents a(n)
height of a triangle
Polygon B represents a(n)
cubic unit
area of a circle
trapezoid
parallelogram
Chapter 12
trapezoid
parallelogram
The arrow in C is pointing to the
.
height of a triangle .
Figure D represents an example of a(n)
πr2 is the formula for the
.
cubic unit
area of a circle
.
.
1
−
(b1 + b2)h is the formula for the area of a trapezoid .
2
203
Glencoe Geometry
Lesson 12-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Active Vocabulary
NAME
DATE
PERIOD
Lesson 12-5 (continued)
Main Idea
Volume of Pyramids
p. 857
Details
Use the pyramid shown to answer the following
questions.
1. What is the height of the pyramid? 9 ft
2. In the formula for finding the volume
1
Bh, what does
of a pyramid, V = −
3
11 ft
9 ft
the B represent?
the pyramid’s base
3. Given that B = 17.5 ft2, find the volume of this pyramid
to the nearest tenth of a cubic foot.
52.5 ft3
Volume of Cones
p. 858
Use the diagram of a cone to fill in the missing
measurements in the chart. Round your answer to the
nearest tenth, as needed.
Volume
cone height
h = 19 cm
h = 19 cm
r = 6.6 cm
V = 866.7 cm3
radius
r = 6.6 cm
Helping You Remember
Many students find it easier to remember
mathematical formulas if they can put them in words. Use words to describe in one
sentence how to find the volume of any pyramid or cone.
Sample answer: Multiply the area of the base by the height and divide that
—————————————————————————————————————————
product by 3.
—————————————————————————————————————————
—————————————————————————————————————————
—————————————————————————————————————————
Chapter 12
204
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1 2
πr h
V=−
3
NAME
12-6
DATE
PERIOD
Surface Areas and Volumes of Spheres
What You’ll Learn
Scan the text in Lesson 12-6. Write two facts you learned
about surface areas and volumes of spheres.
1.
Sample
answer: what it means to be a great circle
____________________________________________________
____________________________________________________
2.
Sample
answer: the formula for finding the
____________________________________________________
volume
of a sphere
____________________________________________________
____________________________________________________
Active Vocabulary
New Vocabulary Label the parts of the sphere pictured below.
chord
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
radius
diameter
tangent
Fill in each blank with the correct term or phrase.
great circle
pole
hemisphere
Chapter 12
One of the two congruent halves of a sphere separated by
hemisphere
the great circle is known as a(n) ___________________.
When a plane intersects a sphere to form a circle which
contains the center of the sphere, this forms
great circle
a(n) ___________________.
pole
A(n) ___________________
is one of the endpoints of a
diameter of a great circle.
205
Glencoe Geometry
Lesson 12-6
____________________________________________________
NAME
DATE
PERIOD
Lesson 12-6 (continued)
Main Idea
Surface Area of Spheres
pp. 864–865
Details
Follow the steps of the flowchart to find the surface
area of the hemisphere.
The hemisphere’s surface area is
comprised of half of the surface
area of the sphere, plus the area of
the great circle.
r = 5 cm
Numerically, that is
1
(4π r2) + π r2.
S=−
2
Using r = 5 cm, find the value, S, of
the surface area of this hemisphere.
S = 235.5 cm2
Volume of Spheres
pp. 866–867
4
πr3 (the formula for finding the
Use the formula V = −
3
volume of a sphere) to answer each question. Round
your answers to the nearest tenth.
2.1 ft3
2. Find the volume of a sphere with a great circle
circumference of 19 inches. 115.8 in3
3. Find the volume of a hemisphere with a radius of
7 centimeters. 718.4 cm3
Helping You Remember
Many students have trouble remembering all of
the formulas they have learned in this chapter. What is an easy way to remember the
formula for the surface area of a sphere?
Sample answer: A sphere does not have any lateral faces or bases, so the
—————————————————————————————————————————
expression in the formula for its surface area has just one term, 4π r2, rather
—————————————————————————————————————————
than being the sum of the expressions for the lateral area and the area of the
—————————————————————————————————————————
bases like the others.
—————————————————————————————————————————
Chapter 12
206
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. Find the volume of a sphere with a diameter of 1.6 feet.
NAME
12-7
DATE
PERIOD
Spherical Geometry
What You’ll Learn
Scan Lesson 12-7. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: Geometry on a Sphere
____________________________________________________
____________________________________________________
2.
Sample
answer: Comparing Euclidean and
____________________________________________________
Active Vocabulary
oblique cone
right cone
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
cross section
isometric view
regular pyramid
slant height
Review Vocabulary Match each term with its definition.
(Lessons 12-1 and 12-3)
the intersection of a solid and a plane
a figure with a base that is a regular polygon and the
altitude has an endpoint at the center of the base
The axis of a cone is also its altitude.
a corner view of a three-dimensional geometric solid on twodimensional paper
the height of each lateral face of a regular pyramid
The axis of a cone is not its altitude.
New Vocabulary Fill in each blank with the correct term or
phrase.
Spherical geometry
is geometry where a plane is
the surface of a sphere.
A geometry where a plane is a flat surface made up of points
that extend infinitely in all directions is known as
Euclidean geometry .
A type of geometry in which at least one of the postulates is
from Euclidean geometry is known as
non-Euclidean geometry .
Chapter 12
207
Glencoe Geometry
Lesson 12-7
Spherical
Geometries
____________________________________________________
NAME
DATE
PERIOD
Lesson 12-7 (continued)
Main Idea
Geometry on a Sphere
pp. 873–874
Details
Use the diagram of the sphere below to answer each
question.
Sample answers:
B
1. Name two lines containing point P.
A
Q
P
!# and BE
!#
FC
C
F
2. Name a segment containing point Q.
−−
BP
E
D
3. Name a triangle.
%BAF
Compare Plane
Euclidean and Spherical
Geometries
p. 874
Complete the table below which compares different
facts about Euclidean geometry and Spherical
geometry.
Euclidean
A plane is a flat surface.
Spherical
The shortest distance
The shortest distance
between two points is a line between two points is a(n)
segment.
arc of a great circle.
Chapter 12
A line is infinite.
A line is finite.
Through any two given
points, there is exactly one
line passing through them.
Through any two given
points, there is one great
circle passing through them.
A unit used to measure
length is a(n) centimeter.
Degrees are the unit used to
measure length.
Two points can be any
distance apart. There is no
greatest distance.
The greatest distance
between two points is 180°.
208
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A plane is a spherical
surface.
NAME
12-8
DATE
PERIOD
Congruent and Similar Solids
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in this lesson.
1.
Sample
answer: how to identify congruent or
____________________________________________________
similar
solids
____________________________________________________
____________________________________________________
2.
Sample
answer: how to use the properties of
____________________________________________________
similar
solids
____________________________________________________
____________________________________________________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
similar solids
New Vocabulary Define the following terms in your own
words.
Sample answer: solids that have exactly the same
————————————————————————————
shape, but not necessarily the same size
————————————————————————————
congruent solids
Sample answer: solids that have exactly the same
————————————————————————————
shape and the same size
————————————————————————————
In your own words, what is meant by scale factor?
Sample answer: Scale factor is the common ratio
————————————————————————————
between the corresponding linear measures of
————————————————————————————
similar solids.
————————————————————————————
————————————————————————————
————————————————————————————
Chapter 12
209
Glencoe Geometry
Lesson 12-8
Active Vocabulary
NAME
DATE
PERIOD
Lesson 12-8 (continued)
Main Idea
Identify Congruent or
Similar Solids
pp. 880–881
Details
Fill in the missing parts to the organizer below.
They have the same shape.
How do you know if
solids are similar?
Ratios of corresponding
linear measures are equal.
Corresponding faces
are ".
Corresponding !
are ".
How to
Identify Congruent
Solids
Corresponding edges
are ".
Apply what you learned with Theorem 12.1 in the
Student Edition to answer the following:
Two similar pyramids have volumes of 125 cubic inches and
64 cubic inches. What is the ratio of the slant height of the
large pyramid to the slant height of the small pyramid?
5:4
Helping You Remember
A good way to remember a new mathematical
concept is to relate it to something you already know. How can what you know about the
units used to measure lengths, areas, and volumes help you to remember the theorem
about the ratios of surface areas and volumes of similar solids?
Sample answer: Lengths are measured in linear units, surface areas in square
—————————————————————————————————————————
units, and volumes in cubic units. Take the scale factor, which is the ratio of
—————————————————————————————————————————
linear measurements in the solids, and square it to get the ratio of their
—————————————————————————————————————————
surface areas or cube it to get the ratio of their volumes.
—————————————————————————————————————————
Chapter 12
210
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Properties of Congruent
and Similar Solids
pp. 881–882
Volumes are equal.
NAME
DATE
PERIOD
Extending Surface Area and Volume
Tie It Together
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Complete the graphic organizer with a formula from the chapter.
Shape
Lateral Area
Surface Area
Volume
Prism
L = Ph
S = L + 2B
V = Bh
Cylinder
L = 2πrh
S = L + 2πr2
V = Bh
Pyramid
1
L=−
Pℓ
S=L+B
1
V=−
Bh
Cone
L = πrℓ
S = L + πr2
1 2
V=−
πr h
S = 4πr2
4 3
V=−
πr
2
Sphere
Chapter 12
211
3
3
3
Glencoe Geometry
NAME
DATE
PERIOD
Extending Surface Area and Volume
Before the Test
Review the ideas you listed in the table at the beginning of the chapter. Cross out any
incorrect information in the first column. Then complete the table by filling in the third
column.
K
W
L
What I know…
What I want to find out…
What I learned…
See students’ work.
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 12 Study Guide and Review in the textbook.
□ I took the Chapter 12 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• Use flash cards to study for tests by writing the concept on one side of the card and its
definition on the other.
Chapter 12
212
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 12.
NAME
DATE
PERIOD
Probability and Measurement
Before You Read
Before you read the chapter, respond to these statements.
1. Write an A if you agree with the statement.
2. Write a D if you disagree with the statement.
Before You Read
Probability and Measurement
See students’ work.
• There are many different ways to
represent the sample space of an
experiment.
• The Fundamental Counting Principle
uses addition to count the total number
of outcomes.
• In a permutation, the order is not
important.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
• Compound events can be independent
or dependent.
• When one event affects another event,
the events are mutually exclusive.
Study Organizer Construct the Foldable as directed at the beginning of this
chapter.
Note Taking Tips
• When taking notes, use a table to make comparisons about the new material.
Determine what will be compared, decide what standards will be used, and then use
what is known to find similarities and differences.
• When taking notes on statistics, include your own statistical examples as you
write down concepts and definitions.
This will help you to better understand statistics.
Chapter 13
213
Glencoe Geometry
NAME
DATE
PERIOD
Probability and Measurement
Key Points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
For example, in the lesson on geometric probability, one fact might be probability that
involves a geometric measure such as length or area is called geometric probability. After
completing the chapter, you can use this table to review for your chapter test.
Lesson
13-1 Representing Sample Spaces
Fact
See students’ work.
13-2 Probabilities with Permutations and
Combinations
13-3 Geometric Probability
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
13-4 Simulations
13-5 Probabilities of Independent and
Dependent Events
13-6 Probabilities of Mutually Exclusive
Events
Chapter 13
214
Glencoe Geometry
NAME
DATE
13-1
PERIOD
Representing Sample Spaces
What You’ll Learn
Scan the text under the Now heading. List two things you
will learn about in this lesson.
1.
Sample
answer: how to use lists, tables, and tree
____________________________________________________
2.
Sample
answer: how to use the Fundamental
____________________________________________________
Counting
Principle to count outcomes
____________________________________________________
Active Vocabulary
sample space
one way of representing a sample space, using line segments
known as branches, to display possible outcomes
tree diagram
experiments with more than two stages
two-stage experiment
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
New Vocabulary Match each term with its definition.
multi-stage experiment
Fundamental Counting
Principle
the set of all possible outcomes of an experiment
a means used to find the entire sample space from an
experiment that does not require listing all the possible
outcomes
an experiment with two stages or events
Vocabulary Link Experiment is a word that is used in
everyday English. Find the definition of experiment using a
dictionary. Explain how its English definition can help you
understand the meaning of experiment in mathematics.
Sample answer: In everyday English, the most
————————————————————————————
common definition of experiment is a test or trial. In
————————————————————————————
mathematics, an experiment involves testing a
————————————————————————————
specific situation when chance, likelihood, or odds
————————————————————————————
are involved.
————————————————————————————
Chapter 13
215
Glencoe Geometry
Lesson 13-1
diagrams
to represent sample spaces
____________________________________________________
NAME
DATE
PERIOD
Lesson 13-1 (continued)
Main Idea
Details
A number cube is rolled twice. Complete the sample
space for this experiment.
Represent a Sample
Space
pp. 899–900
0VUDPNFT
'JSTUSPMM
36
How many possible outcomes are there? ____________
Fundamental Counting
Principle
p. 901
Sara takes orders at a restaurant where each
sandwich is customized. Sara’s manager requires her
to ask the following series of questions when placing
the orders.
Sandwich Shop
Use the Fundamental Counting Principle to find the number
of different types of sandwiches that represent the sample
space for chicken sandwich orders?
16 different types of sandwiches
Chapter 13
216
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. Would you like your chicken fried
or grilled?
2. Would you like bbq, honey mustard,
mayo, or no sauce?
3. Would you like your sandwich
deluxe or regular?
NAME
DATE
13-2
PERIOD
Probability with Permutations and
Combinations
What You’ll Learn
Scan Lesson 13-2. List two headings you would use to make
an outline of this lesson.
1.
Sample
answer: Probability Using Permutations
____________________________________________________
____________________________________________________
2.
Sample
answer: Probability Using Combinations
____________________________________________________
Active Vocabulary
Non-Euclidean geometry
Review Vocabulary Define each term in your own words.
(Lesson 12-7)
a type of geometry in which at least one
————————————————————————————
of the postulates from Euclidean geometry fails
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
————————————————————————————
Euclidean geometry
a geometry where a plane is a flat surface made up
————————————————————————————
of points that extend infinitely in all directions
————————————————————————————
Spherical geometry
a geometry where a plane is the surface of a sphere
————————————————————————————
————————————————————————————
New Vocabulary Write the correct term next to each
definition.
factorial
________________________
given a positive integer n, it is the product of the integers
less than or equal to n
permutation
________________________
an arrangement of objects in which order is important
combination
________________________
an arrangement of objects in which order is not important
circular permutation
________________________
an arrangement of objects in the form of a circle or loop
Chapter 13
217
Glencoe Geometry
Lesson 13-2
____________________________________________________
NAME
DATE
PERIOD
Lesson 13-2 (continued)
Main Idea
Probability Using
Permutations
pp. 906–909
Details
Follow and complete the steps in the flowchart below.
A marching band is divided into squads of 8 musicians.
Each squad is required to select leaders: a head squad leader
and an assistant squad leader. Anders and Matthew are in
one of the squads.
If the positions are decided at random, what is the
probability that Anders and Matthew are selected as
leaders?
Step 1 The number of possible outcomes in the number
of permutations of 8 people taken 2 at a time, 8 P2.
8
8!
8 · 7 · 6!
= 56
P2 = −
= −
(8 - 2)!
6!
Step 2 The number of favorable outcomes in the number
of permutations of these 2 students in each of the 2-leader
positions is 2!, which is 2 · 1 or 2.
Step 3 So, the probability of Anders and Matthew being
56
Probability Using
Combinations
pp. 909–910
28
Leah is packing for a trip to Florida. She decides to
pack 3 of her 8 travel games to take on the trip. If she
chooses to select games at random, what is the
probability that the games chosen are Leah’s
3 favorites?
n!
Use the formula for combinations, nCr = −
, when you
have n objects taken r at a time.
(n - r)!r!
8·7·6·5·4·3·2·1
8!
C3 = −
= −−
=
8
(8 - 3)!3!
(5!)3!
8·7·6·5·4·3·2·1
8·7·6·5·4·3·2·1
−−
= −−
= 56
(5 · 4 · 3 · 2 · 1)(3 · 2 · 1)
(5 · 4 · 3 · 2 · 1)(3 · 2 · 1)
There is only one favorable outcome that the games
————————————————————————————
chosen are Leah’s favorites. So, the probability of
————————————————————————————
1
these 3 games being chosen is −
.
56
————————————————————————————
Chapter 13
218
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
1 .
selected as the 2 squad leaders is −
or −
NAME
DATE
13-3
PERIOD
Geometric Probability
What You’ll Learn
Scan the text in Lesson 13-3. Write two facts you learned
about geometric probability as you scanned the text.
1.
Sample
answer: There are many real-world
____________________________________________________
applications
of probability.
____________________________________________________
____________________________________________________
2.
Sample
answer: Probabilities can be found with
____________________________________________________
length
and with area.
____________________________________________________
Active Vocabulary
Review Vocabulary Fill in each blank with the correct term or
phrase. (Lesson 11-1)
base of a parallelogram
The height of a triangle is the length of an altitude drawn
to a given base.
base of a triangle
height of a parallelogram
height of a triangle
Any side of a triangle is known as the base of a triangle .
Any side of a parallelogram is known as the
base of a parallelogram
.
The height of a parallelogram is the perpendicular
distance between any two parallel bases.
New Vocabulary Define geometric probability in your own
words.
Sample answer: Geometric probability is probability
————————————————————————————
that involves a geometric measure such as length or
————————————————————————————
area.
————————————————————————————
————————————————————————————
————————————————————————————
Chapter 13
219
Glencoe Geometry
Lesson 13-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
____________________________________________________
NAME
DATE
PERIOD
Lesson 13-3 (continued)
Main Idea
Details
Probability with Length
pp. 915–916
The schedule for Meghan’s school is shown on the
number line. The local fire department is going to
randomly have a fire drill today. Find the probability
the fire drill will be during Meghan’s lunch.
2nd bell
8:15-9:05
Homeroom
7:00-7:15
1st bell
7:20-8:10
4th bell
10:05-10:55
3rd bell
9:10-10:00
5th bell
11:35-1:30
lunch
10:55-11:30
dismissal
1:30
probability of fire drill during lunch
Find the total time of
the school day. (Include
time between classes.)
390 minutes
Find the total time of
the lunch period.
35 minutes
Probability with Area
pp. 916–917
Darcy has a magnetic dartboard. What
is the probability that Darcy’s
magnetic darts will land in the central
circle?
5 in.
9 in.
Sample answer: The probability is found by dividing
————————————————————————————
the area of the center circle by the area of the entire
————————————————————————————
π · 52
25π
dartboard: −
=−
≈ 31%.
2
81π
π· 9
————————————————————————————
————————————————————————————
Chapter 13
220
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the probability the drill will
be during lunch.
35 minutes
−
≈ 9%
390 minutes
NAME
DATE
13-4
PERIOD
Simulations
What You’ll Learn
Skim Lesson 13-4. Predict two things that you expect to
learn based on the headings and the Key Concept box.
1.
Sample
answer: how to design a simulation and
____________________________________________________
how
to perform all the steps it involves
____________________________________________________
2.
Sample
answer: how to use different methods for
____________________________________________________
summarizing
and displaying simulation data
____________________________________________________
probability model
simulation
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
random variable
expected value
Law of Large Numbers
New Vocabulary Match each term with its definition.
the average value of a random variable that one anticipates
after repeating an experiment or simulation a theoretically
infinite number of times
a variable that can assume a set of values, each with fixed
probabilities
a mathematical model used to match a random phenomenon
as the number of trials of a random process increases, the
average will approach the expected value
the use of a probability model to recreate a situation again
and again so that the likelihood of various outcomes can be
estimated
Vocabulary Link Simulation is a word that is used in
everyday English. Explain how the English definition can
help you remember how simulation is used in mathematics.
Sample answer: something anticipated or in testing;
————————————————————————————
This is primarily the same as its mathematical
————————————————————————————
definition as it is applied in the study of probability.
————————————————————————————
————————————————————————————
Chapter 13
221
Glencoe Geometry
Lesson 13-4
Active Vocabulary
NAME
DATE
PERIOD
Lesson 13-4 (continued)
Main Idea
Design a Simulation
pp. 923–924
Details
Miranda bowled a strike in 75% of her frames over the
last 9 months. Design a simulation that can be used to
estimate the probability that she will bowl a strike in
the next frame.
Complete the missing pieces to the steps below.
Step 1
Possible Outcomes
She bowls a strike.
She does not bowl a strike.
theoretical probability: 25%
theoretical probability: 75%
Step 2 Assume that Miranda will have the opportunity to
bowl 50 more frames.
Step 4 A trial, one spin of a spinner, will represent bowling
1 frame.
What will a successful trial represent?
strike
What will a failed trial represent?
not a strike
The simulation will consist of how many spins? 50
Summarize Data from
a Simulation
pp. 924–926
Conduct the simulation above and record your results
in the table provided below. When your simulation is
complete, find the experimental probability of
Miranda bowling a strike on her next frame.
Outcome
Tally
strike
11
not a strike
39
Total
50
number of strikes
−−
=
number of non-strikes
Chapter 13
Frequency
222
11
−
39
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Step 3 Divide the spinner provided into
two sectors to represent the two
possible outcomes listed in Step 1.
bowl a strike
not bowl a strike
75%(360°)
25%(360°)
NAME
13-5
DATE
PERIOD
Probabilities of Independent and Dependent
Events
What You’ll Learn
Skim the Examples in Lesson 13-5. Predict two things you
think you will learn about this lesson.
1.
Sample
answer: how to identify independent and
____________________________________________________
dependent
events
____________________________________________________
2.
Sample
answer: how to find the probability of
____________________________________________________
independent
events
____________________________________________________
Active Vocabulary
a specific type of tree diagram that includes probabilities
independent events
________________________
events where the probability of one event occurring does not
affect the probability of the other event occurring
compound event
________________________
consists of two or more simple events
conditional probability
________________________
a probability that uses the notation P(B!A)
dependent events
________________________
events where the probability of one event occurring in some
way changes the probability that the other event occurs
Vocabulary Link Independent and dependent are words used
in everyday English. How do their English definitions apply
to their definitions in probability?
Sample answer: Dependent means relying on
————————————————————————————
something. In the case of probabilities, dependent
————————————————————————————
means one event’s probability does rely on another
————————————————————————————
event’s probability. Independent means not influenced
————————————————————————————
by others. In the case of probabilities, independent
————————————————————————————
events have no influence on each other’s probabilities.
————————————————————————————
Chapter 13
223
Glencoe Geometry
Lesson 13-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
probability tree
________________________
New Vocabulary Write the correct term next to each
definition.
NAME
DATE
PERIOD
Lesson 13-5 (continued)
Main Idea
Independent and
Dependent Events
pp. 931–933
Details
The probability that two independents occur follows
the multiplication rule for probability: P(A and B) =
P(A) · P(B). Extend the use of this multiplication rule
for probability by creating a model of another set of
independent events that can be used to demonstrate
this rule. Sample answer:
Suppose a number cube is rolled and a coin is
tossed.
The sample space for this experiment is: {(1, H), (2, H),
(3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T),
(5, T), (6, T)}.
Using the sample space, the probability of the
compound event of rolling a 6 and the coin landing on
1
.
heads is P(6 and H) = −
12
The same probability can be found by multiplying the
multiplication rule for probability). P(6) = −16 ; P(H) = −12 ;
1
P(6 and H) = −16 · −12 or −
12
Conditional
Probabilities
p. 934
Ms. Alexander’s class of 16 students sits at four
different tables, with four students at each table.
Ms. Alexander chose tables A and B to go to the lab,
while those students at tables C and D remained
seated. If Carly has to go to the lab, then what is the
probability that she sits at Table A?
Sample answer: Because Carly is going to the lab, the
————————————————————————————
sample space is reduced from the entire class, to just
————————————————————————————
the 8 students seated at tables A and B. So, the
————————————————————————————
4
1
probability that Carly is at Table A is P(A| lab) = −
or −
.
8
2
———————————————————————————
Chapter 13
224
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
probabilities of each simple event (hence, using the
NAME
DATE
13-6
PERIOD
Probabilities of Mutually Exclusive Events
What You’ll Learn
Skim the lesson. Write two things you already know about
mutually exclusive events.
1.
Sample
answer: Some events affect each other,
____________________________________________________
2.
Sample
answer: The probability that an event will
____________________________________________________
not
occur is called the complement.
____________________________________________________
Active Vocabulary
Review Vocabulary Label the part of each diagram to which
the arrow is pointing. (Lessons 11-1 and 11-2)
height of a triangle
trapezoid
base of a parallelogram
trapezoid
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
height of a triangle
height of a parallelogram
3. base of a parallelogram
4. height of a parallelogram
New Vocabulary Define the following terms in your own
words.
mutually exclusive events
Sample answer: two events that cannot happen at the
————————————————————————————
same time and have no common outcomes
————————————————————————————
complement
Sample answer: the probability that an event will not
————————————————————————————
occur; This is found by subtracting the probability
————————————————————————————
that the event will occur from 1.
————————————————————————————
————————————————————————————
Chapter 13
225
Glencoe Geometry
Lesson 13-6
while
others do not affect each other.
____________________________________________________
NAME
DATE
PERIOD
Lesson 13-6 (continued)
Main Idea
Mutually Exclusive
Events
pp. 938–940
Details
Model the following mutually exclusive events by
representing their relationship in a Venn diagram
below.
When a die is rolled, find the probability of rolling a 1 or 6.
Is the following event mutually exclusive or not mutually
exclusive?
when a die is rolled, the probability of rolling a number
less than 5 or an odd number not mutually exclusive
At the local carnival, the probability of a dart to land
on one of the playing cards on the board is 33%.
Name the complement of landing on one of the cards and the
probability of the complement occurring.
Sample answer: The complement is the dart not
————————————————————————————
landing on a playing card. The probability of this
————————————————————————————
happening is 67%.
————————————————————————————
————————————————————————————
Chapter 13
226
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Probabilities of
Complements
pp. 941–942
NAME
DATE
PERIOD
Probability and Measurement
Tie It Together
Complete each graphic organizer with a definition or formula from the chapter.
Sample Space
Experiment
Outcome
Event
a situation involving chance that leads to results
the result of a single trial of an experiment
one or more outcomes of an experiment
Calculating Probability
Fundamental Counting
Principle
Permutations
n
n!
Pr = −
Combinations
n
n!
Cr = −
Circular Permutations
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The total possible outcomes in an experiment with k
stages is n1 • n2 • n3 • … • nk.
(n - r)!
(n - r)!r!
n!
−
n
Simulations
the use of a probability model to recreate a situation and
estimate the probability of various outcomes
Complement
P(not A) = 1 - P(A)
Types of Probability
Geometric Probability
Independent Events
Dependent Events
Conditional Probability
Mutually Exclusive
Events
Chapter 13
probability that involves a geometric measure like length
or area
P(A and B) = P(A) • P(B)
P(A and B) = P(A) • P(B|A)
P(A and B)
P(A)
P(B|A) = −
P(A and B) = P(A) + P(B)
227
Glencoe Geometry
NAME
DATE
PERIOD
Probability and Measurement
Before the Test
Now that you have read and worked through the chapter, think about what you have
learned and complete the table below. Compare your previous answers with these.
1. Write an A if you agree with the statement.
2. Write a D if you disagree with the statement.
Probability and Measurement
• There are many different ways to represent the sample
space of an experiment.
• The Fundamental Counting Principle uses addition to
count the total number of outcomes.
• In a permutation, the order is not important.
• Compound events can be independent or dependent.
• When one event affects another event the events are
mutually exclusive.
After You Read
A
D
D
A
D
Are You Ready for the Chapter Test?
Use this checklist to help you study.
□ I used my Foldable to complete the review of all or most lessons.
□ I completed the Chapter 13 Study Guide and Review in the textbook.
□ I took the Chapter 13 Practice Test in the textbook.
□ I used the online resources for additional review options.
□ I reviewed my homework assignments and made corrections to incorrect problems.
□ I reviewed all vocabulary from the chapter and their definitions.
Study Tips
• You will do better on a test if you are relaxed. If you feel anxious, try some deep
breathing exercises. Don’t worry about how quickly others are finishing; do your best
and use all the time that is available to you.
Chapter 13
228
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes,
personal tutors, and practice tests to help you study for concepts in Chapter 13.