Download Geometry - Chapter 1 Day #1 - Somerset Independent Schools

1
Geometry - Chapter 1 Day #1
Topic: Introduction to Geometry
PART I. Basic facts to know about geometry
1.
Geometry - the study of the measure of the Earth.
The word is derived from two Greek words:
"Geo "Meaning Earth and "metria" meaning 'to measure.'
2. The most famous among all Greeks was, Euclid.
a. He was a mathematician and teacher.
b. He combined the geometric knowledge of his time with a better and more logical
system. He didn't invent geometry, he just organized it.
e. The newer, improved system of geometry is known as: Euclidean Geometry.
PART II. Terms to know:
1. Conditional--A statement formed by an "IF-THEN" where the "If' is the hypothesis and
"Then" is the conclusion.
a. Doesn't have to be math related.
b. These are always true.
c. Example: If it is raining, then I will carry an umbrella.
2. Converse -- A statement formed by switching the hypothesis and the conclusion of a
conditional.
3.
4.
a. These are not always true.
b. Example: If I carry an umbrella, then it is raining.
Definition -- A statement that gives meaning to a term.
a. The "textbook" or "dictionary meaning of a term, word, or concept."
Postulate -- A statement that is accepted as truth without proof.
a.
DOES NOT have to be proven.
b. It is assumed to be true.
5. Theorem -- A statement that must be proven before being accepted as truth.
a. These are the absolute law.
b. No proving necessary.
PART III. Key Points to Remember
1. Converses of definitions are a/ways true.
2. Converses of postulates and theorems mayor may not be true.
Examples:
Answer each question:
1.
If the 'if part of a conditional
is called the
2.
The 'then' part of a conditional
3.
When the hypothesis and the conclusion of a conditional are interchanged,
statement is the
of the conditional.
is called the
_
_
the resulting
Give the hypothesis and the conclusion of each statement.
1.
If you are funny, then you will make people laugh.
2.
If a triangle has all sides equal, then the triangle must be equilateral.
3.
If a nonzero whole number has exactly two factors, then the number is prime.
2
Write a conditional with the given hypothesis and conclusion.
1.
HYPOTHESIS: A man lives in Houston.
CONCLUSION: He lives in Texas.
2.
HYPOTHESIS: A number is prime.
CONCLUSION: The number has exactly two divisors.
Write the converse of each statement.
1.
If a person is swimming, then that person is wet.
2.
If a-b
= 0, then a = 0 or b = O.
Write each statement as a conditional. Then determine whether it is true or false.
1.
A number ends in five if it is even.
2.
All left-handed
3.
'm' ending in 0, 2,4, 6, or 8 implies that 'm' is an even number.
people are male.
Finally, today we will look at something known as a 'Venn Diagram.'
Venn diagram - is used to illustrate how a set of things that satisfy the hypothesis lies inside the
set of things that satisfy the conclusion.
Write a conditional statement that each Venn diagram illustrates.
#1.
#2.
#3.
Draw a Venn diagram to illustrate each statement.
1.
If an angle measures 100 degrees, then it is obtuse.
2.
If you are the captain of your team, then you are a junior or a senior.
3.
Peace Corps volunteers want to help other people.
Write three separate conditional
statements
that the Venn diagram below shows:
3
Homework Chapter 1 Day #1
(Separate Sheet of Paper)
I.
Give the hypothesis and the conclusion of each statement.
1.
2.
3.
4.
5.
II.
If Michele's age is 15, then she is too young to vote ..
If a point is in the exterior of an angle, then it cannot be in its interior.
If a number ends in a five, then it is an odd number.
If Jim is 18 years old, then he may vote in the election.
If a number is positive, then it is greater than zero.
Write a conditional with the given hypothesis and conclusion.
1. HYPOTHESIS:A woman lives in Chicago.
CONCLUSION: She lives in Illinois.
2. HYPOTHESIS:A triangle has no sides equal.
CONCLUSION:The triangle is scalene.
3. HYPOTHESIS:It rains before noon.
CONCLUSION: We will not play tennis.
4. HYPOTHESIS:An angle's measure is 90 degrees.
CONCLUSION:The angle is a right angle.
5. HYPOTHESIS:It is raining.
CONCLUSION: We will play chess.
III.
Write the converse of each statement.
1.
2.
3.
4.
5.
Write each statement as a conditional. Then determine whether it is true or
false.
IV.
1.
2.
3.
4.
5.
V.
#1.
VI.
If you are 21 years old or older, then you can legally vote ..
If an angle is an acute angle, then the angle measures less than 90 degrees.
If a triangle is equilateral, then its sides are all equal.
If the temperature outside is below freezing, then ice can form on the sidewalks.
If a figure is a rectangle, then it has exactly four sides.
Go to law school and you will become a judge.
An angle is obtuse only if it has measure less than 90 degrees ..
'q' ending in 1, 3, 5, 7, or 9 implies that 'q' is an odd number.
You will fail the course only if you get an F on the final exam.
A catfish is a fish that has no scales.
Write a conditional statement that each Venn diagram illustrates.
#2.
Draw a Venn diagram to illustrate each statement.
1. If a figure is a square, then it is a rectangle.
2. If the game is rugby, then the game is a team sport.
3. If a figure is a rhombus, then it is a quadrilateral.
4
VII.
Multiple choice
1. What is the hypothesis of the given statement?
"lf pigs had wings, you could flv."
a. Pigs have wings.
b. You can fly.
c. Pigs do not have wings.
d. You cannot fly.
2. Which statement is the converse of the given statement?
"lf you make an insurance claim, then your rates will go up.
a. If your insurance rates do not go up, then you have not made a claim.
b. If you do not make an insurance claim, then your rates will not go up.
c. If your insurance rates go up, then you have made an insurance claim.
d. If you make an insurance claim, then your rates will not go up.
3. Which
a.
b.
c.
d.
conditional and its converse are BOTH true?
If x = 1, then 2x =2
If x = 3, then X2 = 6.
If x = 2, then X2 = 4.
If X2 = 4, then x = 2.
4.
of the following best defines a postulate?
A statement accepted without proof
A conclusion reached using inductive reasoning.
An example that proves a theorem false.
A statement that you CAN prove true.
5.
Which
a.
b.
c.
d.
What is the converse of the following statement?
"If a whole number has a as its last digit, then the number is evenly divisible by 10./1
a. If a number is evenly divisible by la, then it is a whole number.
b. If a whole number is divisible by la, then it is an even number.
c. If a whole number is evenly divisible by la, then it has a on its last digit.
d. If a whole number has 0 as its last digit, then it must be evenly divisible by
10.
1
Geometry - Chapter 1 Day #2
Topic: Points, Lines, and Planes
PART I. Points. Lines. and Planes: DEFINITIONS
1. Point - A point indicates position, but is has no size (is represented
2.
Line - is a set of all points that extend infinitely
3.
Plane - surface that extends without
4.
Intersection - Is the set of points that two figures have in common.
5.
Collinear Points - Points that are on the same line.
6.
Non-collinear
7.
Coplanar Points - Points that are on the same plane.
8.
Non-Coplanar - Points that are NOT on the same plane.
9.
Space - is a bound-less, 3-dimensional
by a dot).
in two opposite directions.
end in all directions.
Points - Points that are not on the same line.
set of points.
PART II. Postulates about points. lines, and planes:
Postulate - is a statement that is accepted as truth without
1.
proof.
2.
A line, plane, and space each contain an infinite number of points. Some points in a plane
are non-collinear. Some points in space are non-coplanar.
Through any two points, there is exactly one line.
3.
Through any three non-collinear
4.
If two points lie in a plane, then the line joining them lies in the plane.
5.
If two planes intersect, then their intersection
6.
If two lines intersect, then their intersection
7.
If a plane and a line intersect, then their intersection
points, there is exactly one plane.
is a line.
is a point.
is a point.
Examples:
Answer the following:
names for this line:
n .•.
~':::':"'~.""'_-041.~---4."'-"''''''
p
e j::
x.. '<;
1.
Give five different
2.
Two lines that have a point in common are called
lines.
2
3.
Three points that are all on a line are
points.
4.
Four points that are not in the same plane are
points.
Refer to the figure to answer the following questions:
1.
Name a line that contains points T and P.
2.
Name a line that intersects the plane containing points 0, N, and P.
3.
Name the plane that containsTN
&
QR.
Draw and label a figure for each relationship described below:
1.
2.
3.
4.
AK& CG intersect
at point M in plane T.
Two points M and R inside of plane Z.
Plane B contains line q.
Three non-collinear points: R, 5, and T.
5. Point L between points Sand T
Refer to the figure to answer the following.
T
1.
How many planes are shown in the figure?
2.
Name three collinear points.
3.
Are points N, R, 5, and W coplanar?
Q
Explain.
Refer to the figure to answer the following questions.
0 and
1.
Name a line that contains points
N.
2.
Name a point between K and P.
3.
Why is point L not between points K and P?
4.
Name four sets of three collinear points.
p
Name the geometric term(s) modeled by each object.
1.
A stop sign
2.
The tip of a needle or pin
3.
The strings on a fiddle
4.
A car antenna
5.
A library card
3
HOMEWORK Worksheet
Chapter 1 Day #2
Answer the following:
I.
1. Give five different names for this line: ~
It..
A
." (..,
L
G
2. True or False (explain false): Planes and lines contain an infinite number of
points.
3. True or False (explain false): A line can be cut in half.
For #'s 4-6, determine if the statement is true or false, based on the figure below:
4. The intersection of lines RTand UV is at point 5. ,.'.
5. Points R, 5, and T are coplanar.
~
6. V is between Rand T.
II.
T
Refer to the figure to answer the following questions:
1.
2.
3.
4.
III.
Name a line that contains point D.
Name a point contained in line n.
What is another name for line p?
Name the plane containing lines nand p.
Draw and label a figure for each relationship described below:
1.
2.
3.
4.
5.
6.
IV.
Gii
inside of plane Q
Two points J and K inside of plane R.
Point K lies on line RT
Plane J contains line s
Line YP lies in Plane B and contains point C, but does not contain point H.
Lines q and f intersect at point Z in plane U.
Refer to the figure to answer the following.
1.
2.
3.
4.
V.
How many planes are shown in the figure?
How many of the planes contain points F and E?
Name four points that are coplanar
Are points A, B, and (coplanar? Explain.
Refer to the figure to answer the following questions.
1. Name a line that contains points A and B.
2. Name a point between A and C.
3. Why is point F not between points A and C?
Name a pair of lines that intersect at point A.
5. Name a point NOT on line AC.
4.
D •
4
VI.
1.
2.
Multiple Choice:
terrrn is 'undefined'?
Which geometric
a.
Segment
b.
c.
Collinear
Ray
d.
Plane
Look at the figure shown.
a.
Line AD
b.
Line CB
c.
Line CD
d.
Line BF
3. What is the smallest
a. 2
b. 3
c.
4
d.
Undefined
Where
do planes ACE and BCD intersect?
8
D--F'"
number
of distinct
4. At how many points can two distinct
a. 1
b. 2
c.
points that can define
lines intersect?
3
d. 4
5. In the figure shown below, which line is the same as Ei5
a.
b.
c.
d.
ML
DM
Nl
MN
6. If two lines are coplanar, which of the following
7.
a.
The lines intersect
b.
The lines never intersect
c.
All points on the lines are coplanar
d.
The lines share at least one point.
What is the intersection
a.
A point
b.
A line
c.
A line segment
d.
A ray
A
of two distinct,
must be true?
non-parallel
planes?
a plane?
1
Geometry - Chapter 1 Day #3
Linear Measure & Precision
Terms to know:
1. Line segment - The set of all points between 2 endpoints.
2. Congruent - means 'equal measure'. SYMBOL:
_
3. Congruent Segments - Two segments that have the same length.
4. Between ness for points - A point is between 2 other points if:
a. The 3 points are collinear AND
b. Segment length + Segment length = Whole Segment. (Segment
Addition Postulate)
EXAMPLE: If XY + Yl = Xl, which of the points X, Y, or Z is between the other two?
Topic:
I.
5. Precision - depends on the smallest unit available on measuring tool.
EXAMPLES:
a. Find the precision measurement of 5 mm.
b. Find the precision measurement of 6.5 cm.
c. Find the precision measurement of 8.25 km.
d. Find the precision measurement of 10 inches.
e. Find the precision measurement of 8 Yz inches.
f.
Find the precision measurement of 7 }:i inches.
Examples
II.
A.
#1.
Find the length of each line segment or object.
•
•
I'~~""~'" r""" r ""'' 1' "'I' "I' "I" "I
'I '"
I·
in.
1
B. Find the value of the variable and KL if K is between J and L.
#1. JK = 6r, KL = 3r, and JL = 27
#2. JK
= 2s, KL ;;; s + 2, and
JL = S5 -10
I
2
2
C. Q is the midpoint of P R. What are po, QR, and PR?
.-
D. Find the measure of each segment below:
.
#1. PS
16 tf
.,~
p
1l.7~
Q
#2. AD
~ ~"
• -x•
#3. WX
&.V
(
..----~~----~.~-~~--.
.P
I
••
A
S
E. On a number line, the coordinates of P, 0, R, and 5 are -12, -5, 0, and 7,
respectively.
a. Draw a sketch of this number line.
b. Using the sketch, which line segment is the shortest?
c. Using the sketch, which line segment is the longest?
d. Which line segments are congruent?
F. Use the figures to determine whether each pair of segments is congruent.
#1. TU, SW
#2. AD, Be
#3. GF, FE
T,
2ft
S
2~n3ft
~w
)
--
12.7 in.
GD
\
5X
H
12.9 in.
--- --
F
6x
E
y
3
HOMEWORK Chapter 1 Day #3
I.
Measurements
Find the length of each fine segment or object.
#1.
P
#2.
Q
•
r-ITT·l-p-·-I-rrrTTT·rTTI···-fl
! 0
L__._.
...
1
c
•
•
...
D
•
j-pTnllmplTlllmpTflTmfllTfillTITITnTITmrmrl
[rnrn
L_.
2 i
1
2
3
. .,..
I
j
#3.
4
i
5
I
~ ~
#4.
Segments & Between-ness
II.
Find the value of the variable and LM if L is between Nand M.
= 5x, LM = 3x, and NL = 40. Find x and
NL = 6x - 5, LM = 2x + 3, and NM = 30.
1. NL
2.
III.
LM.
Number lines & Segment Measures
Use the number line at the right to answer the following questions:
1. If R5 = 15 and 5T = 9, then RT = ?
2. If 5T = 15 and RT = 40, then RS = ?
•••~---3.
R5
= 8y + 4, ST = 4y + 8, and
RT = 1Sy - 9.
a. What is the value of y?
PJ
••
T
Find RS, ST, and RT.
b.
IV.
..•••
,....---
Precision
Find the precision for each measurement.
Find the measure and the interval.
1. 80 inches
2. 16 ~ meters
3. 3 X feet
V.
Segment lengths.
Find the measurement
#1. AC
of each segment shown.
#2.
ST
R 1.2
A 16.7 mm B 12.8 mm
C
#3.
em
S
~40;m~
T
XY
'I-.
••••
- __
3t/
t
4
VI.
Congruent
Segments.
Use the figures to determine
whether each pair of segments
#2. EF, FG
#1. AB, CD
is congruent or not.
#3. N P, LM
E
F
VII.
~
N
1.75 in.
0.75~
8ft
G
P
7
M 0.75 in. L
Midpoints
A is the midpoint of segment XY.
1.
Find the value of x
•
2. Find XA
3. Find AY
4. Find XY
VIII.
•
•
Number lines
On a number line, the coordinates of X, Y, Z, and Ware -7, -3, 1, and 5,
respectively.
1.
Draw a sketch of this number line
2. Find the lengths of segments XY and ZW. Are they congruent?
3. Find the lengths of segments ZX and WY. Are they congruent?
4. Find the lengths of segments ZY and XW. Are they congruent?
IX.
Multiple Choice
1. A basketball coach is interested
ability.
in measuring a player's vertical jumping
While standing and reaching up, the player's hand reaches the 4
inch mark on the scale. When jumping,
the player can reach the 19 inch
mark. What is the player's vertical jumping
a.
ability?
19 inches
b. 4 inches
c. 23 inches
d. 15inches
2.
If PQ + QR
= PR, which
of the points P, Q, or R is between the other two?
a. P
b. Q
c.
R
d. Can't be determined
from the information
given.
Points X, V, and l are collinear and Y is between X and Z. Which
statement
must be true?
a. XY = YZ
b. XZ-XY = YZ
c. XY + Xl Yl
=
d. XZ
= XY -
YZ
1
Geometry - Chapter 1 Day #4
Topic: Distance & Midpoints
Terms to Know
I.
1. Ruler Postulate - To every point on a line, there corresponds exactly one real
number.
2. Segment Addition Postulate - The name for the "2nd part" ofthe definition of
"between ness" for points.
Ex: AB + BC AC; if B is between A &
=
c.
(part + part
= whole)
DISTANCE
./ Can occur in the coordinate plane, as well as on a number line .
./ Distance must always be a positive value.
Distance on the Number line:
3.
E ,:. b #.,
I ~I
~'-~~-'~~'-~-'-+'
•.
+I-'i-P'-t
•
£,
A
~
~
b
-::,-- ..- 'f
p
C
-s
,l
-;) -I
Cr
h
5"
I &. .~ ij
EXAMPLES:
Give the coordinate of each of the following:
#1. Point G
#2. Point F
#3. Point A
Name the point with the given coordinate.
#2. 6
#3. -4 ~
#1. -1
Find the following distances.
#1. AD
#2. BG
#3. CG
-~-,-r-y
Distance on the Coordinate Plane:
'--
-
-f----
-
M
#2. E
#3. Z
#4. S
#2. ES
Use the distance formula to find the distance between each pair of points.
#1. L (-7, 0), Y (5, 9)
#2. U (1,3), B (4, 6)
. -,
S
-I-r,;~
--lot'
-
-~---
- ---c-II' -
Use the distance formula to find the distance between each pair of points.
#1. MZ
-
-.. -.- '- -
EXAMPLES:
Give the caordina tes for the following points:
#1.
f--- -
f--
2
4. MIDPOINT - The point halfway between the endpoints of a segment .
./ Can occur in the coordinate plane, as well as on a number line .
./ Midpoints can be negative answers .
.,f
Midpoint Theorem -If
M is a midpoint of AB, then AM = X AB and BM =
XAB .
./
Segment Bisector - segment, line, or plane that intersects a segment at
its midpoint.
Finding Midpoints
on the Number Line:
P
R
Q
.,+"+",+,,+,,+,,.
-10 -8 -€
--4
-2
S
0
T
2
4
6
EXAMPLES:
Use the number line to find the coordinate of the midpoint of each segment.
#1. RT
#2. QR
#3. ST
#4. PR
Finding Midpoints
on the Coordinate Plane:
Find the coordinates of the midpoint of a segment having the given endpoints.
#1. K (-9, 3), H (5, 7)
#2. W (-12, -7), T (-8, -4)
Find the coordinates of the missing endpoints given that E is the midpoint of DF.
#1. F (5, 8), E (4, 3)
Free Response Question: The midpoint of segment AB is on the y-axis, and
segment AB is parallel to the x-axis. Point A is located in Quadrant III. Which
quadrant contains point B? Explain.
3
HOMEWORK Chapter 1 Day #4
I.
Use the number line to find each measure.
Be
A
0
E
F
1. DE
.\ , \ \ , , \ , I , I I , I I I'
2.
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
CF
3. AB
II.
Use the distance formula to find the distance between each pair of points.
1.
A (0; 0); B(8, 6)
2. S(-2, -1); T (6,5)
3. Refer to the figure below and find the distance between the two points:
~.
y
-
IR(1,5L
!
1
I I
I
y
I
1
/ P(3.4L
/
I
0(-5,3)
1
y
I
I
~
i--""""I
I --
~
I
71
x
0
x
0
N(-2,-2)
'Iii
----'~-
III.
I
I
I
Use the number line to find the coordinate of the midpoint of each segment.
-
1
A
1. Segment AC
2. Segment DF
3. Segment CE
<I +,
"'7
c.
9
~tll
"'S
..:;
e
F
llt-t-t-+I~
~
-I.
E
D
l
;
:,
7
Cf
IV.
Find the coordinates of a segment having the given endpoints.
1. A(8, 4); B (12, 2)
2. G(4, 2); H(8, -6)
v.
Find the coordinates of the missing endpoint given that S is the midpoint of
segment RT.
1. T(-4, 3); 5 (-1,5)
2. T(2, 8); S(-2, 2)
VI.
MULTIPLE CHOICE
1. What is the other ENDPOINT of the segment with MIDPOINT -3 and
ENDPOINT -77
a. -11
b.
c.
-5
1
d. 4
2. According to the graph, what is the MIDPOINT of segment AB?
a. (1, 0)
b. (1, -O.S)
c. (1, 0.5)
d. 1.5, -0.5)
y
B
/
1
/
x
0
~
A
V
4
Point M is the midpoint of segment AB. What is the value of x in the figure?
y
a. 1.5
B(2X,2X)
A
b. 5
M(7,8)
c. 5.5
A(4,6)
d. 11
a
4.
The midpoint
a. Xl
b.
Xl
c. Yl
d.
5.
Yl
of segment Xl is point Y. Which of the following
x
is true?
= XY
= Y2 XY
= Y2 XY
= Y2 Xl
The diagram shows distance in meters.
How far, in meters, is it from the
parking lot to your house?
VII.
a.
44m
b.
c.
183 m
d.
189 m
135m
•
I
tm•
-27
(!J
•
I
0
I
)
162
Free Response Questions
1.
The room shown below is 14 ft by 10 ft. Find the dimensions
furniture
to the nearest tenth.
10
y
-/
8
2.
a.
Length and width of the dresser
b.
Length and width of the table.
c.
Length and width of the table
The midpoint
r7
r7
4
'" r-, ~
o
of segment AB is in Quadrant
i/
1/
6
2
of each piece of
./
f"
:'\.1/
"
1/
KV
2
I'\.
l'\
et
"17
1/
/IT.u
"r\.:'\.
V
4
1
1/
1
re
6
8
10
51
r
I
12
x
14
IV, and segment AB is parallel
to the y-axis.
a.
What quadrant
b.
What else can you determine
or quadrants
cannot contain either point A or B? Explain
about points A and B?
1
Geometry - Chapter 1 Day #5
Topic: Angle Measure
I.
Definitions & Examples
1.
Rays - A part of a line; has 1 endpoint and extends infinitely
in 1 direction.
2.
Angles - formed by 2 non-collinear
3.
Vertex - A common endpoint of 2 rays that make up an angle.
4.
Right Angle - An angle whose measure is
5.
Acute Angle - An angle whose measure is,
6.
Obtuse Angle - An angle whose measure is
7.
Straight Angle - An angle whose measure is
8.
Opposite Rays - Two rays with a common endpoint that are
9.
Congruent Angles - Angles with the same measure
rays with a common endpoint.
_
_
_
_
10. Angle Bisector - A ray that cuts an angle into 2 congruent angles.
11.
Angle Addition Postulate
_
2
II.
Short Answer
Use the figure to name each af the following
1. A right angle, an obtuse angle, an acute angle
2.
A point in the interior of <EBC & a point in the exterior of <EBA
D.
3. The angle bisector of <ESC
4. A point on <CBE
5.
The sides of <ABF
6.
A pair of
7.
The common
~-------~-~----~e-~
G
B
A
opposite rays
vertex of all angles shown in the figure.
8. A pair of congruent angles
9. The angle with the greatest measure.
III.
Examples
Use the figure at the right to answer questions 1-6:
1.
Name the vertex of each angle.
4-5,4-3,4-8,4-NMP
2.
Name the sides of each angle.
4-6, 4-2, 4-MOP, 4-0MN
3. Name a point in the interior of 4NMP.
4.
Name a point in the exterior of 4-NOM.
5. Write another name for 4-QPR.
6.
Write another name for 4-1.
7. Classify each angle as right, acute, or obtuse.
a. <UZW
b.
<YZW
C.
<TZW
d.
<UZT
x
T
z
y
3
HOMEWORK - Chapter 1 Day #5
I.
Matching
Match each description below with its appropriate term. Some terms may be used more than once or
not at al/.
1.
2.
3.
A figure made up of two non-collinear rays with a common endpoint.
Angles whose degree measures are less than 90.
Angles that have the same measure
Angles whose degree measure are between 90 and 180 degrees
4.
5. A tool used to measure angles
6. The common endpoint of the rays that form an angle.
7. A ray that divides an angle into two congruent angles.
WORD BANK:
Vertex
Angle
Right angles
Angle bisector
Obtuse angles
Acute angles
Opposite rays
Congruent angles
Compass
Protractor
II.
Short Answer
Refer to the figure.
1.
2.
3.
4.
III.
Name
Name
Name
Write
the vertex of <4.
the vertex of <2.
the sides of <BOC.
another name for <OBC.
Multiple
A8
_
of2Y\
.
c
Choice
Refer to the figure for the next TWO questions:
1. Which of the following pairs are acute angles?
a. <1 and <2
b. <2 and <5
c. <2 and <3
d. <4 and <1
2.
SV
bisects <RSV. If m<RSV
a. 32
b. 64
c. 116
= 64, what
is m<RST?
d. 128
3.
Using the diagram shown at the right, which of the conclusions can you make from the
information in the diagram?
a.
MN =LM
b.
<MNL
c.
m<MNL = 2m<LMN
d.
LM
== <LMN
= 2 MN
L
N
4
IV. REVIEW: Error Analysis
For each exercise, identify the error(s} in planning the solution or solving the problem. Then write the
correct solution.
#1. Use the diagram.
ANSWER:
What are the names of the ray with point D as its endpoint?
AD, BiJ, CD
#2. What is the intersection
of plane ABC and plane CDH?
B~
I
,,
ANSWER: point C
A r---''----/
I
D
,
I
r----
__
/ F
/
/
E
"
--------Y:
H
#3. B is the midpoint of segment AC. What is AC?
ANSWER/SOLUTION:
3x+2 = 6x-13
-sx = 15
x=5
~3x+2~~6
A
•
B
x
.
C
1
Geometry - Chapter 1 Day #6
Topic: Angle Measure Part 2
I.
Use the diagram below for
Exercises1-9.
Find the measure of each angle.
1. LADB
2. LFDE
3. LBDC
4. LCDE
o
A
5.LADC
6. LFDC
7.LBDE
8. LADE
F
9.LBDF
II.
Examples
1.
AC bisects
2.
In the figure, CB and
are opposite rays, CE bisects <DCF, and
a. If m<DCE = 4x + 15 and m<ECF = 6x - 5, find m<DCE.
<DAB. If m<DAC = 58, what is m<DAB?
CD
CG bisects
<FCB.
E
o
c
B
b.
If m<FCG = 9x + 3 and m<GCB
= 13x -
9, find m<GCB.
2
3.
In the figure below.
a.
BA
and
Be are
opposite ravs. jjj} bisects <EBe. and
BF bisects
If m<EBD = 4x + 16 and m<DBC :: 6x + 4, find m<EBD.
b.
If m<ABF
= 7x -
8 and m<EBF :: 5x
+ 10 , fInd m<EBF.
4 •
::
A- ~
4.
<ABE.
X is in the interior of LUN. mLlIN
x?
== 100 m
a.
What is the value of
b.
What are mLLlX and mLXIN?
'
-
LUX - 14t, and mLXIN
8
==
_-_-J;;
t + 10.
5. Z is in the interior of LGHI. mLGHI == 170, mLGHZ == 3 - 5
a. What is the value of s?
s
, and mLZHI == 2s + 25.
b. What are mLGHZ and mLZHI?
.
3
HOMEWORK - Chapter 1 Day #6
Use the diagram below for Exercises 1-11. Find the measure of each angle.
I.
1. LMLN
2. LNLP
3. LNLQ
4. LOLP
5. LMLQ
M
Q
Classify each angle as acute, right, obtuse, or straight.
6. LMLN
7. LNLO
8. LMLP
9. LOLP
10. LOLQ
11. LMLQ
Use the figure below for Exercises 12 and 13.
w
1
y
x
2
z
12. What is another name for LXYW?
13. What is another name for LWYZ?
II.
Problems
QS bisects
QP
QR
<PQT, and
and
are opposite rays.
1. If m<PQT = 60 and m<PQS 4x + 14, find the value of x.
=
p
2.
If m<PQS
= 3x + 13 and m<SQT = 6x - 2, find m<PQT.
BA and BC are opposite
3.
If <EBF = 6x
4.
If m<l
rays,
BF
bisects <CBE, and
Bi5 bisects
<ABE.
+ 4 and m<CBF = 7x - 2, find m<EBC.
= 4x + 10 and
m<2
= 5x, find
A
m<2.
REVIEW:
Write true or false.
1. A, D, F are coplanar.
~
+-+
2. AC and FE are coplanar.
E
3. A, 8, E are coplanar.
4. D, A, 8, E are coplanar.
B
5. A and C are collinear.
6. D, E, and 8 are collinear.
#7. How many sets of four collinear points are there in a 4-by-4 geoboard as pictured at the right?
•
•
•
•
• • •
• • •
• • •
• • •
1
Geometry - Chapter 1 Day #7
Topic: Angle Relationships
I.
1.
Important Concepts& terms
Adjacent Angles - angles in the same plane that have a common vertex and a common side,
but no common interior points.
2.
Vertical Angles - are two nonadjacent angles formed by two intersecting
lines.
3.
Linear Pair - A pair of adjacent angles whose non-common
4.
Complementary
Angles - Two angles whose measures have a sum of 90°
5.
Supplementary
Angles - Two angles whose measures have a sum of 180
6.
Perpendicular
Lines - Lines, rays or segments that form four right angles.
7.
Angle Addition Postulate - (similar to the Segment Addition Postulate.
8.
9.
Protractor Postulate - Every angle measures between 0 degrees and 180 degrees.
Vertical Angle Theorem - If two angles are vertical angles, then they are congruent.
sides are opposite rays.
0
•
10. Linear Pair Postulate -If two angles form a linear pair, then they are also supplementary.
II.
Examples
1.
Identify each pair of angles as adjacent, vertical, complementary,
and/or a linear pair.
a. <1 & <2
b. <1 & <4
c.
<3 & <4
d. <4 & <5
e.
<1 & <3
supplementary,
~1
2
2.
If <1 & <2 form a linear pair and m<2 :::115°, then find m<l.
3.
If <1 & <2 are complementary
4.
If m<14 = 4x - 26 and m<3 = 3x + 4, find x.
5.
Find the value of x in the following:
~. 7)(+10
6.
.,.~
••_~_
••••
r...-__
---j>"".
and m<2 = 20°, find m<l.
fox.
b.
c.
For the figure below, find the value of x and determine
if AB 1. CD.
C--------~----------D
7.
8.
Use the figure at the right to answer the following:
a.
If m<FGE
= 5x + 10, find
b.
If m<BGC
= 16x -
x so
thatFC
4 and m<CGD
.1
AE.
= 2x + 13, find
x so that <BGD is a right angle.
Determine whether each statement can be assumed from the figure.
a.
<NQO and <OQP are complementary.
b.
<SRQ and <QRP is a linear pair.
c.
<MQN and <MQR are vertical angles.
Explai
3
HOMEWORK Chapter 1 Day #7
I.
Short Answer
1. Identify each pair of angles as adjacent, vertical,
complementary, supplementary, and/or a linear pair.
a. <1 & <2
b. <1 & <6
c. <1 & <5
d. <3 & <2
v
R
~ p.
- -----
2. Determine whether each statement can be assumed from the figure. Explain.V'
a. <WZU is a right angle
W
b. <YZUand <UZV are supplementary
X.
c. <VZU is adjacent to <YZX.
II.
Problems
1. If <1 & <2 form a linear pair and m<2 = 5r, then find m<l.
'A
2. If <5 & <6 are complementary and m<5
= 22°, find
m<6.
3. Find the value of x in the following:
a.
4. If <11
b.
= 11x and m<12 = lOx +10, find x.
c
5. For the figure below, find the value of x and determine if AB .1 CD.
D
4
6. Use the figure at the right to answer the following:
----+
p
.....-.
a. If m<RTS= 8x + 18, find x so that TR .1 TS.
b. If m<PTQ = 3y - 10 and m<QTR = y, find y so that <PTRis a right angle.
III.
Multiple Choice
1. What is m<BAC?
a. 25
b. 50
c. 130
d. 155
2. What
a.
b.
c.
d.
is another name for <4?
<VWS
<SVW
<SWV
<WVT
s
A
3. m<KLM = 129 and m<MNO = 129. What is true about these two angles?
a. They are both acute angles.
b. They are congruent
c. They are both right angles.
d. They are both straight angles.
m<MRT = 133. What is m<MRN?
a. 24
b. 48
c. 46
d. 87
-*4.
5.
N
M
t2g - 2)°
(4g ~ 9)°
R
T
<UB and <IJM are congruent. If the sum of the measures of the angles is 90,
what type of angle are they?
a. Acute
b. Obtuse
c. Right
d. Straight
1
Geometry - Chapter 1 Day #8
Topic: Constructions
Today, we are going to learn how to use a straightedge and a compass to construct various figures
such as: segments, angles, perpendicular bisectors, and angle bisectors.
We are able to use the tools just mentioned (compass and straightedge) to make afigure that is
congruent to an original figure without actually having to measure it. This method is more accurate
than merely sketching and drawing it.
Vocabulary:
1.
Straightedge
2.
Compass - a geometric
- is a ruler with no markings on it.
3.
Construction
tool used to draw circles and parts of circles called arcs.
- Is a geometric
Example #1: We will construct
figure drawn using a straightedge
congruent
segments.
Construct a segment congruent to a given segment.
Given: AB
Construct:
CD so that CD == AB.
Step 1: Draw a ray with endpoint
and a compass.
•
A
B
C.
Step 2: Open the compass to the length of AB.
Step 3: With the same compass setting, put the compass point on point C. Draw and arc that
intersects
the ray. Label the point of intersection
Extension of Example #1: Use a straightedge
Example #2: We will construct
congruent
D.
to draw XY. Then construct
RS so that RS = 2·XY.
angles.
Construct an angle congruent to a given angle.
Given:
<A
Construct:
<5 so that <5
==
<A
Step 1: Draw a ray with endpoint
S.
Step 2: With the compass point on vertex A, draw an arc that intersects
point of intersection
the sides of <A. label the
Band C.
Step 3: With the same compass setting, put the compass point on point S. Draw an arc and label its
point of intersection
with the ray as R.
Step 4: Open the compass to the length Be. Keeping the same compass setting, put the compass
point on R. Draw an arc to locate point T.
Step 5: Draw
Sf
Extension of Example #2: Construct
<F so that m<F
= 2·m<B.
2
~
Vocabulary:
4.
Perpendicular
lines - are two lines that intersect to form right angles.
s.
Perpendicular
bisector - is a line, segment, or ray that is perpendicular
~
to the segment at its
midpoint.
Example #3: We will construct a perpendicular bisector.
Construct the perpendicular bisector of a segment.
Given:
A
B
AB
Construct:
Xv
so that
Xv
is the perpendicular
AB.
bisector of
Step 1: Put the compass point on point A and draw a long arc as shown.
Be sure the opening is
greater than ~ AB.
Step 2: With the same compass setting, put the compass point on point B and draw another long
arc. Label the points where the two arcs intersect as X and Y.
Step 3: Draw
XV
XY.
Label the point of intersection
1. Ai} at midpoint
M, so
XV
of AB and
is the perpendicular
Extension of Example #3: DrawST.
Construct its perpendicular
Construct the bisector of an angle.
Given: <A
A
AD, the
bisector of <A.
as M, the midpoint of AB.
bisector of AB.
Example #4: We will construct the bisector of an angle.
Construct:
XY
bisector.
<:
_
Step 1: Put the compass point on vertex A. Draw an arc that intersects the sides of <A. Label the
points of intersection Band C.
Step 2: Put the compass point on point C and draw an arc. With the same compass setting, draw
an arc using point B. Be sure the arcs intersect.
Label the points where the two arcs intersect at D.
---->
Step 3: Draw AD
Extension of Example #4: Draw obtuse <XYZ. Then, construct its bisector
YP.
3
Additional
Examples:
Draw a diagram similar to the given one. Then do the construction. Check your work with a ruler or
a protractor.
1.
Construct
XY congruent to AB.
2.
Construct
VW so that VW
= 2·AB.
A
3.
Construct
DE so that DE
4.
Construct
QJ
5.
Construct
<D so that <D
6.
Construct
<F so that m<F
7.
Construct
the perpendicular
8.
Construct
<Z so that m<Z
so that OJ
= TR
+ PS.
= TR -
PS.
==
•
T
<c.
= 2·m<C.
= )12
bisector of AB.
m<X.
(see figure)
xL----~
9.
Construct a triangle
(see figure)
~
B
whose sides are all the same length as M N.
.. ----------------~.H
R
P
5
4
HOMEWORK Chapter 1 Day #8
Do the constructions using the figures be/ow. Check your work with a ruler or a protractor.
I.
Constructions
1. Construct AB congruent to XY.
2. Construct the perpendicular bisector of XY.
3. Construct a triangle whose sides are all the same length as XV.
•
y
•
X
4. Construct AB so that AB = MN + OP.
5. Construct KL so that Kl OP - MN.
=
6. Construct the perpendicular bisector of MN. M----N
7. Construct the perpendicular bisector of OP.
8. Construct <A so that m<A
= m<l
o
•
p
+ m<2.
9. Construct <Cso that m<C = 2·m<2.
II.
10. Construct <Y so that m<Y = Yz m<2.
Extensions
1. Use a ruler to draw segments of 2 ern, 4 ern, and 5 cm. Then construct each triangle with
the given side measures, if possible. If it is not possible, explain why not.
a. 4 em, 4 em, and 5 cm
b. 2 ern, 5 ern, and 5 cm
e. 2 ern, 2 em, and 5 cm
d. 2 em, 2 em, and 4 em
2.
M is the midpoint of XY. Find the value of x. Show YO\Jrwork. (HINT: use either factoring
or quadratic formula).
...1 ?_
•
III.
"
J...
.
~
..•
Multiple Choice
Which of the following materials does NOT have to be used when making basic
constructions?
a. A ruler
b. A compass
c. A straightedge
d. A pencil
2. Which of the following must be true about a perpendicular bisector and the segment it
bisects?
a. The perpendicular bisector and the segment bisect each other.
b. The angle of intersection depends on the length of the line segment.
c. The perpendicular bisector intersects the segment at a 45 degree angle.
d. The perpendicular bisector intersects the segment at a 90 degree angle.
1.
1
Geometry - Chapter 1 Day #9
Topic: Polygons
I.
Important Concepts & Terms
1. Polygons - is a closed figure formed by a
number of coplanar line
segments.
v' Must have at least 3 sides (otherwise, not a closed figure)
./ Polygons are named based on the number of their sides:
NAME OF POLYGON
n=3
n=4
n=5
n=6
n=7
n=8
n=9
n = 10
n = 12
n = 11, 13 & #'5 higher
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
'n'-gon
2. Regular Polygon - Is a polygon with congruent sides and congruent angles.
3. Irregular
A convex polygon in which all the sides are
EXAMPLES OF POLYGONS:
congruent and all the angles are congruent is
1. Convex
called a regular polygon. Octagon PQRSTUVW
below is a regular octagon.
vOs
2. Concave
U
T
P
Q
W
R
Perimeter - is the sum of the lengths of all the sides of the polygon .
./ Special formulas for certain polygons:
TRIANGLE
SQUARE
RECTANGLE
2
EXAMPLES:
I.
Name each polygon by its number of sides. Then
classify it as concave
or
#3.t:J
#2'0
convex AND as being regular or irregular.
#6
II.
#1.
Find the perimeter of each figure.
#2.
)CmA3Cm
5.5ft
(f
D
2~
square
3.5cm
#3.~
#4.
1CN
7Yd
(S
19yd
12yd
14yd
24yd
#5. What is the effect on the perimeter of the figure in #1, if each measure is
multiplied by 57
#6. What is the effect on the perimeter of the figure in #2 if each measure is divided
by 37
III.
Find the length of each side of the polygon for the given perimeter.
#1. P = 104 millimeters
#2. P =: 48
o
#3. P =: 96
J(
2x
rectangle
#4. An 8 foot by 10 foot rug leaves 1 foot of the bedroom floor exposed on all four
sides. Find the perimeter of the bedroom floor.
IV.
Coordinate Geometry
Graph each figure in the coordinate plane. Find the perimeter.
#1. X(-4, 2), Y(2, 10), Z{2, 2)
#2. A(O, 0), B(O, 5), C(6, 5), D(6, 0)
3
HOMEWORK - Chapter 1 Day #9
I.
Classifyingpolygons
Name each polygon by its number of sides. Then classify it as concove or convex.
Also, classify it as either being regular or irregular.
#0
#1 .:
#3.
o
#6.
caution or warning
L
Identify the shape of each traffic sign.
#4.
#5.
school zone
yield
cID
II.
Perimeter
Find the perimeter of each figure.
#1.
28 ft
#20
~
1
4
#3.
12m
13ft
13 ftl
VI
a
4
I
7
15m
28 It
6m
4
15m
8
#4. What is the effect on the perimeter of the figure in #1 above, if each measure is
multiplied by 4?
#5. What is the effect on the perimeter of the figure in #2 above, if each measure is
tripled?
#6. What is the effect on the perimeter of the figure in #3 above, if each measure is
divided by 2?
Find the length of each side of the polygon for the given perimeter.
#7. P = 90 centimeters
MO
G
H
L
:
K
J
#8. P = 31 units.
r---z
#9. P = 42 inches
3n + 2
'II
X-1~R
T
3x- 5
II
4
Graph each figure in the coordinate plane. Find the perimeter.
#10. R(l, 2}, 5(1, -2), T(4, -2)
#11. L(-3, 2), M(2, ~4), N(2, 20), P(-3, 20)
#12. A garden that is 5 ft by 6 ft has a walkway 2 ft wide around it. What is the
amount of fencing needed to surround the walkway?
'*
III.
Multiple Choice
Choose the correct letter.
*
1. A 12 ft by 15 ft swimming pool has a 3 ft wide no slip surface around it. What
is the outer perimeter of the no-slip surface?
a. 78 ft
b. 78 ft2
c. 198 ft
d. 198 ft2
2. What is the perimeter of ~PQR with vertices P(-2, 9), Q(7, -3) and R (-2, -3) in
the coordinate plane?
a. 21 units
b. 25 units
c. 34 units
d. 36 units
IV.
Free Response
1. The units of the floor plan at the right are in feet.
Find the perimeter
room.
a. The kitchen
rlt( nen
.
b. The bedroom
of each
~.!Ol01: m
I
c. The bathroom
20 ft
IV al
h~1I va
d. The closet
C os at
Batt robn I
I .
20 ft
2.
Sometimes, students confuse 'area' with 'perimeter'.
remodeling
a house and you needed to determine
buy. Tell whether
below.
a. Wallpaper
Suppose you were
how much of each item to
you need to know AREA or PERIMETER for each situation
for a bedroom
b. Crown molding for a ceiling
c. Fencing for a backyard
d. Paint for a basement floor.