Download Section 2.1 – Undefined terms, postulates, segments and angles

Mth 97
Fall 2013
Chapters 1 and 2
Problem Solving (Chapter 1)
Polya’s 4 Steps: 1) Understand the problem;
2) Devise a plan;
3) Carry out the plan;
4) Look Back
Problem Solving Strategies
1. Draw a Picture (See pages 5 and 6 of your text)
– Work problem 2 on page 10.
2. Guess and Test (pages 6 – 7)
– Work problem 17 on page 12 using toothpicks.
3. Use a Variable (pages 8 – 9) – Work the following problem.
The perimeter of a picture is 26 inches. Its length is 3 inches greater than its width. What are the
dimensions of the picture? (Hint: Define a variable, then write and solve an equation.)
Section 2.1 – Undefined terms, postulates, segments and angles
Axiomatic System: Undefined terms → Definitions → Postulates → Theorems
Undefined terms: Can be described but cannot be given precise definitions
Point – suggested a
Line – suggested by a
Space: The set of ________ points
Plane – suggested by a
Geometric Figure: Any collection of _____________
Collinear Points: Points that line on the same ____________.
Non-collinear Points: Points that are ________ all on the same line.
Coplanar Points: Points that lie in the same ________________.
Non-coplanar Points: Points that are_______ contained in the same plane.
Postulates: Statements that we assume to be ___________which state relationships between defined
and undefined terms.
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Mth 97
Fall 2013
Chapters 1 and 2
Theorems: Results deduced from undefined terms, definitions or postulates and can be _____________
Relationships Among Points, Lines, and Planes
Postulate 2.1 – Every line contains at least two distinct points.
Postulate 2.2 – Two points are contained in one and only one line.
Postulate 2.3 – If two points are in a plane the line containing these points is also in the plane.
Postulate 2.4 – Three non-collinear points are contained in one and only one plane and every plane
contains at least three non-collinear points.
Postulate 2.5 – In space, there exist at least four points that are not all coplanar.
Line Segment: determined by its endpoints & includes all the points __________ these points on a line
Postulate 2.6 (The Ruler Postulate)
– Every line can be made into an exact copy of the real number line using 1–1 correspondence.
A
B
C
D
E
F
G
H
I
J
-4
-3
-2
-1
0
1
2
3
4
5
The distance between two points is the ___________________ difference between their coordinates.
AE =
BF =
DH =
Congruent segments are segments that have the same______________________.
AB  DE because AB = DE = 1.
CF 
CF 
CF 
because CF = ______ = ______ = _______ = ______
If the coordinates of A, B, and C are respectively 3 5 , 2 5 , and  5 , find each segment length.
AB =
BC =
AC =
A ray is a portion of a line that has one endpoint and extends indefinitely in _________ direction and is
named by the endpoint followed by a point on the ray written below a ray symbol,
A
B
C
D
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Mth 97
Fall 2013
Chapters 1 and 2
Angles and Their Measure
Angle: Formed by two line segments or rays meeting at a common endpoint.
Vertex ______
Sides
_________ and _________
Named with an angle symbol and its vertex _____________
or an angle symbol and three points (side-vertex-other side)
_____________ or ____________
Postulate 2.7 (The Protractor Postulate)
If we place one ray of an angle at 0° on a protractor and we place the vertex at the midpoint of the
bottom edge (or center crosshairs), then there is a 1-1 correspondence between all other rays that can
serve as the second side of the angle and the real numbers between 0° and 180°. This number is called
the “measure” of the angle.
Measure the angle above and then do number 1 on today’s ICA (In-class Assignment).
Types of Angles
Acute– has a measure
Right – has a measure of 90°
between 0° and 90°
Straight – has a measure of 180°
Obtuse – has a measure
between 90° and 180°
Reflex – has a measure between 180° and 360°
Congruent angles have the same ________________________
If the measure of DEF is 30° and DEF  GHI , then the measure of GHI is ____________
Complementary – two angles whose sum is 90°
Supplementary – two angles whose sum is 180°
Do number 2 on today’s ICA.
Adjacent angles – have a common vertex and side but no common interior points.
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Mth 97
Fall 2013
Chapters 1 and 2
mAGB  75
Given
mCGE  90
C
B
mCGD  50
mFGC  175
mDGE 
Find
A
mFGE 
D
G
mAGF 
mBGC 
In the drawing above name pairs of:
Complementary angles
E
F
Supplementary angles
Adjacent angles
Dimensional Analysis is used to make unit conversions.
Review of Fractions
4 5

5 6
4 5

5 6
We use these two ideas along with unit fractions (fractions equal to 1) to do dimensional analysis.
12 in = 1 ft
1 hr = 60 min
If you know how two units of measure are related, you can write a unit fraction. Instead of figuring
out whether you want to multiply or divide by the relationship between two units of measure, we will
always multiply by a unit fraction that allows us to CANCEL LABELS to achieve the desired unit of
measure.
More precise measures of angles include fractions of degrees or minutes and seconds, where
60 minutes = 1 degree (60’ = 1o) and 60 seconds = I minute (60” = 1’).
Use dimensional analysis to express 32°15’40” in degrees as a decimal rounded to thousandths.
Use dimensional analysis to express 15.12°
in degrees and minutes.
Use dimensional analysis to express 75.3°
in degrees minutes and seconds.
Complete the rest of ICA 1.
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Mth 97
Fall 2013
Chapters 1 and 2
One radian is the measure of the central angle of a circle that cuts off a portion of the distance around
the circle that is the same length as the radius of the circle. The radian measure of a full circle is 2π, so
2π rads = 360° or for half a rotation, π rads = 180°. Use dimensional analysis to find the radian measure
for the following angles.
unit frations
15°
120°
Problem Solving Strategies (cont.)
4. Look for a Pattern (See page 16) Discuss problems 11a and 11b on page 25.
5. Use Make a Table (See pages 18-19) and inductive reasoning to solve problem 3 on page 24.
6. Solve a Simpler Problem (See page 21) Use this strategy to solve problems 17 on page 26.
A
2.2 Polygons (2 dimensional shapes that lie in a plane)
Triangles are determined by three _____________________________ points.
∆ABC is formed by __________ __________ and ___________
B
C
We can classify triangles by their sides and angles.
Definition
No two sides are 
Type
Acute
Definition
All 3 acute angles
Isosceles
At least 2 sides are 
Right
One right angle
Equilateral
All 3 sides are 
Obtuse
One obtuse angle
Equiangular
All 3 angles are 
Angles
Sides
Types
Scalene
Sketch the following triangles:
Isosceles Right ∆
Scalene Obtuse ∆
Scalene Acute ∆
Equilateral ∆
Obtuse Isosceles ∆
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Mth 97
Fall 2013
Chapters 1 and 2
Simple Closed Curves
A simple closed curve is a figure that lies in a plane and can be traced so that the starting and ending
points are the same and no part of the curve is crossed or retraced. Closed curve Not a closed curve
A circle is a simple closed curve that consists of the set of all points ______________________
from a given point, called the center of the circle. A circle is named by its center point.
A Polygon is a simple closed curve composed of line ______________________. Polygons are
named listing the vertices in order around a figure. Adjacent vertices are the endpoints of a
__________ of a polygon. Two sides that share a _______________ are adjacent.
Polygons
Not a polygon
Polygons are usually described by the number of sides they have.
Type of Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Number of Sides
Type of Polygon
Octagon
Decagon
n-gon
Number of Sides
Parallel Lines lie in the same ______________ and do not intersect.
Quadrilaterals are named by the lengths of their sides, the measure of their angles, or some other
attribute such as parallelism.
Quads Square
Def.
Rectangle
Rhombus Parallelogram Trapezoid
All sides are All angles are All sides
right angles. are
 and all
angles are
.
right angles.
Opposite sides Exactly one
are parallel.
pair of
parallel
sides.
Isosceles
Trapezoid
Kite
A trapezoid
whose
non-parallel
sides are  .
Two pairs of
adjacent  sides
where the pairs
have no side in
common.
Pix
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Mth 97
Fall 2013
Chapters 1 and 2
Do problems 1 to 7 of ICA 2.
Types of Symmetry in Polygons
Reflection Symmetry: A figure has reflection symmetry if there is a ______________(axis of symmetry)
along which the figure may be folded so that one half of the figure matches exactly the other half.
Draw and cut out each of the following, then fold them to find all the lines of symmetry. Tell how many
lines of symmetry each polygon has and sketch them on the drawings below.
isosceles triangle
equilateral triangle
square
rectangle
____ lines of symmetry
____ lines of symmetry
____ lines of symmetry
____ lines of symmetry
Rotational Symmetry: a figure has rotational symmetry if it can be rotated about a point ______than a
full turn, so that the image is identical to the original figure.
If the above polygons have rotational symmetry, tell what angle(s) each shape must be rotated to achieve
rotational symmetry.
isosceles triangle
rotational symmetry?
_______
If yes, degree(s) rotated?
____________
equilateral triangle
_______
____________
square
_______
____________
rectangle
_______
____________
2.3 Angle measure in Polygons and Tessellations
Theorem 2.1 – Angle Measure in a Triangle
The sum of the measures of the angles in a triangle is _______
Draw and cut out any triangle, then follow directions on page 74 for problem 35a & 35b.
53°
Find the missing angle measures.
x°
34°
x=
d° f°
65°
a=
d=
b=
e=
c°
a°
124°
eq
uat
ion
her c =
e. f =
e°
b°
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Mth 97
Fall 2013
Chapters 1 and 2
Do the rest of ICA 2
A
E
B
The angles in a polygon are called ______________ angles. Find the sum
of the measures of the vertex angles of a pentagon.
Hint: Draw all possible diagonals from a single vertex count
the number of triangles formed.
D
E
Theorem 2.2 – Angle Measure in a Polygon
The sum of the measures of the vertex angles in a polygon with n sides is (n – 2) 180°.
A Regular Polygon is a polygon in which all the sides are congruent and all the angles are congruent.
Which of the above polygons are regular?
How could we find the measure of a vertex angle in a regular pentagon?
Theorem 2.3 – Vertex Angle Measure in a Regular Polygon
The measure of the vertex angle of a regular n-gon is
Find the measure of the vertex angle
of a regular hexagon.
 n  2 180 .
n
A vertex angle of a regular polygon
measures 157.5°. How many sides does the
polygon have?
Use equations to solve problem 11 on page 72.
a)
b)
c)
d)
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Mth 97
Fall 2013
Chapters 1 and 2
Tessellations
A tessellation is a tile-like pattern formed by repeating _____________to fill a plane without gaps or
overlaps. The shapes are polygonal regions, formed by each polygon together with the portion of the
plane that it encloses. A regular tessellation (p. 70) is composed of regular polygons that are all the same
________and ___________. Which tessellation below is regular?
2.4 Three Dimensional Shapes (Solids)
A Polyhedron is a three-dimensional shape composed of polygonal regions, any two of which have at
most a common side. The plural is polyhedra. It must be enclosed with no ______________.
The polygonal regions are called ____________________
The common line segments are called _____________________
The point where the edges meet is called a __________________
Page 80, problem 2: a)
b)
c)
Types of Polyhedra: Prisms, Pyramids, Regular Polyhedrons
A Prism is a polyhedron with two opposite identical faces, called __________, which are identical
polygonal regions in ________________ planes. The other faces are called _________________ faces.
The height of the prism is the ________________________ distance between the bases.
Prisms are named by the shape of their ________________.
Right Prisms have lateral faces that are _____________________________ regions.
Oblique Prisms have lateral faces that are _______ rectangular regions.
Page 82, problem 10: a)
b)
A Pyramid is a polyhedron consisting of a polygonal region for its ___________ and triangular regions
as _________________ faces, which all meet at a point called the ____________ of the pyramid.
The height is the ___________________ distance between the apex and the base.
Pyramids are named by the shape of their _____________.
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Mth 97
Fall 2013
Chapters 1 and 2
A right regular pyramid’s base is a _______________polygon;
its lateral faces are ___________________ triangles;
and its slant height is the height of any of its ___________________.
An oblique regular pyramid’s lateral faces are NOT isosceles ___________________.
Page 83, problem 13: a) ______________________________________________________
(3 or 4 words)
b) ______________________________________________________
A Regular Polyhedron is a polyhedron in which all the _________are identical, regular polygons, that is,
all the faces are regular polygons of exactly the same size and shape. Only five exist.
Polyhedron
Tetrahedron
Hexahedron
Octahedron
Dodecahedron
Icosahedron
Shape of the face
Triangle
Square
Triangle
Pentagon
Triangle
# of faces (F)
4
6
12
20
#of vertices (V)
4
6
20
12
# of edges (E)
12
12
30
30
Euler’s Formula, F + V = E + 2, holds true for all polyhedra, not just regular polyhedra. Verify Euler’s
Formula for a) Octahedron
b) Square based pyramid
c) Pentagonal prism
Page 82, problem 11:
Do page 1 of ICA 3
Other Solids: Cylinders, Cones, and Spheres
A circular cylinder is formed by two congruent circles in _______________ planes together with the
surface formed by line segments joining corresponding points of the two circles.
The two circular regions are the ___________________
In a right circular cylinder, the segments joining corresponding points of the two bases are
_________________________ to the bases.
In an oblique circular cylinder, the segments joining corresponding points of the two bases
are _______ perpendicular to the bases.
A circular cone is formed by a circular region, called the ___________, together with the surface formed
when the apex (a point not in the base) is joined by line segments to every point on the circle.
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Mth 97
Fall 2013
Chapters 1 and 2
If the apex of a circular cone lies on the line that is perpendicular to the center of the base,
the cone is a _________________ circular cone. Otherwise, the cone is ___________________.
The height of a cone is the _______________________ distance to the base.
The slant height in right circular cones is the distance from the apex to a point on the edge of the
_____________.
A sphere is the set of all points in space that are a fixed distance from a given point, the_____________ of
the sphere. The ____________ is a segment whose endpoints are the center and a point on the sphere.
The ________________ is a segment that contains the center of the sphere and whose endpoints are on
the sphere.
2.5 Dimensional Analysis (Use “Units of Measurement” – the last page of the textbook)
Ex.1: Covert 9 feet to inches.
Ex. 2: Convert 50 pounds to ounces
Ex. 3: A water bottle contains 24 ounces of water.
How many quarts does it hold?
Ex. 4: Convert 21 m into inches.
Ex. 5: π radians = 180°. Use this to convert radians
Ex. 6: How many square inches in 10
to degrees.
square feet?
Ex. 7: Lynn walks for exercise every day. She averages 6km/hr. What is her speed in m/sec?
Do the rest of ICA 3.
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