Unit 7
Unit 7 Lesson 1: Polygons
CCSS
Lesson Goals
G-CO 9: Prove theorems about

lines and angles.
G-CO 10: Prove theorems about
triangles.
G-CO 11: Prove theorems about
parallelograms.
Identify, name, and describe
polygons.
ESLRs: Becoming Effective Communicators,
Competent Learners and Complex Thinkers
Previously in Math: Polygon
a 2-dimensional plane figure
Formed by three or more line segments
Adjacent sides are non-collinear
Each side intersects exactly 2 other sides
at their endpoints
Previously in Math: Polygon
C
D
A
B
E
F
L
K
H
G
J
I
polygon ABCDEFGIJKL
Previously in Math: Polygon
Previously in Math: Classification
hexagon
triangle
quadrilateral
heptagon
pentagon
octagon
Previously in Math: Classification
nonagon
dodecagon
decagon
n-gon
classifications
Convex
No line that contains a side passes through
the interior of the polygon
interior
Classification
Convex
All vertices
“point” outward.
classifications
Convex
No line that contains a side passes through
the interior of the polygon
Concave
A polygon that is not convex
interior
Classification
Concave
At least one vertex
“points” inward.
You Try
Classify by number of sides.
octagon
You Try
Convex or Concave?
concave
You Try
Classify by number of sides.
dodecagon
You Try
Convex or Concave?
convex
You Try
Classify by number of sides.
heptagon
You Try
Convex or Concave?
concave
Previously in Math: Classification
Equilateral: all sides congruent
Equiangular: all angles congruent
Regular: equilateral and equiangular
Polygon Interior Angles Theorem
The sum of the measures of the
interior angles of a convex polygon is
180  n  2  
where n is the number of sides.
m1  m2  m3  m4  180  4  2  
1
2
3
4
example
Find the sum of the measures of the interior angles
of a convex decagon.
180  n  2 
180 10  2 
180  8 
1440
example
Find x.
Sum of the angle measures  180  n  2  
180 8  2    1080
n8
160
125
135
135
125
130
 125
935
x  935  1080
x  145
example
Find x.
Sum of the angle measures  180  n  2  
180  5  2    540
n5
7 x  309  540
112
115
 82
309
7 x  231
x  33
example
Find the value of x.
 x  25 
x  x   x  25   x  25  360
4x  50  360
4x  410
x  102 12
x
 x  25 
x
You Try
Find the measure of each interior angle.
Is the quadrilateral regular?
Not regular
A
x
80
D
 x  20 
x
C
B
x  x   x  20  80  360
3x  60  360
3x  300
x  100
mA  mC  100
mB  80
Summary
A polygon is __________________.
A polygon is classified by _________,
_________, or ________.
A regular polygon has _____________.
The Polygon Interior Angle Sum Theorem says
____________.
p. 325: 2 – 9, 21 – 23, 27 -30, 38, 42 – 46 e
p. 665: 8, 12, 14, 16
p. 325: 2 – 9, 21 – 23, 27 -30, 38, 42 – 46 e
 p. 665: 8, 12, 14, 16

CONVEX OR CONCAVE Use the number of sides to tell what kind of
polygon the shape is. Then state whether the polygon is convex or concave.
pentagon
convex
heptagon
concave
heptagon
concave
RECOGNIZING PROPERTIES State whether the polygon is best described as
equilateral, equiangular, regular, or none of these.
equilateral
regular
equiangular
TRAFFIC SIGNS Use the number of sides of the traffic sign to tell what kind
of polygon it is. Is it equilateral, equiangular, regular, or none of these?
quadrilateral
regular
pentagon
none
triangle
regular
octagon
regular
ANGLE MEASURES Use the information in the diagram to find m A.
m A 55 110 124  360
m A 289  360
m A  71
USING ALGEBRA Use the information in the diagram to solve for x.
3x  90  60 150  360
3x  300  360
3x  60
x  20
USING ALGEBRA Use the information in the diagram to solve for x.
 4x  10  3x  108  67  360
7 x 185  360
7 x  175
x  25
USING ALGEBRA Use the information in the diagram to solve for x.
x 2  90  90  99  360
x  279  360
2
x 2  81
x 9
Unit 7 Lesson 1b: Polygons
CCSS
Lesson Goals
G-CO 9: Prove theorems about

lines and angles.
G-CO 10: Prove theorems about
triangles.
G-CO 11: Prove theorems about
parallelograms.
Identify, name, and describe
polygons.
ESLRs: Becoming Effective Communicators,
Competent Learners and Complex Thinkers
Regular Interior Angle Corollary
The measure of each interior angle
of a regular polygon is
180  n  2 
n
example
Find the measure of one interior angle of
a regular polygon with 15 sides.
180( n  2)
n
180(15  2)
15
180(13)
15
156
example
The measure of each interior angle of
a regular polygon is 165o.
How many sides does the polygon have?
180(n  2)
 165
n
180n  360  165n
360  15n
24  n
Polygon Exterior Angle Theorem
The sum of the measures of the exterior angles of
a convex polygon (one angle at each vertex) is 360o.
6
1
m1  m2  m3  m4
 m5  m6  360
5
2
4
3
Knott’s Berry Farm
A portable ferris
wheel in England
Austria - Vienna:
Riesenrad - the Giant Wheel at the Prater
You Try
What is the sum of the measures of the
interior angles of a decagon?
180  n  2 
180 10  2 
180  8 
1440
You Try
What is the of the measures of one interior
angle of a regular decagon?
180 10
n 22
10
n
1440
10
144
You Try
What is the sum of the measures of the
exterior angles of a decagon?
360
You Try
What is the sum of the measures of the
interior angles of a heptagon?
180  n  2 
180  7  2 
180  5
900
You Try
What is the measures of one interior angle of a
regular heptagon?
180  n7  2 
n7
900
7
900
You Try
What is the sum of the measures of the
exterior angles of a heptagon?
360
You Try
What is the sum of the measures of the
exterior angles of a polygon with 45 sides?
360
You Try
What is the sum of the measures of the
exterior angles of a polygon with 27 sides?
360
example
Find the value of y.
yo
2yo
o
y
2yo
2 y  y  2 y  y  360
6 y  360
y  60
Regular Polygon Exterior Angle
Corollary
The measure of each exterior angles of
a regular polygon
360
n
example
Find the value of x.
xo
360
n
360
6
60
Summary
The Regular Interior Angle Corollary extends the
Polygon Regular Interior Angle Sum Theorem by
__________________.
Polygon Exterior Angle Theorem shows that a
polygon becomes closer to a _____ as the
_____ of sides ______ because ________.
The Regular Polygon Exterior Angle Corollary
extends the Polygon Exterior Angle Theorem by
_____________.
p. 325: 47 – 51
p. 665: 18 – 24 e, 30 – 36 e, 55, 56
 p. 325:
47 – 51
 p. 665: 18 – 24 e, 30 – 36 e, 55, 56
hexagon
convex
regular
octagon
concave
equilateral
pentagon
convex
none
17-gon
concave
none
ANGLE MEASURES In Exercises 14–19, find the value of x.
180  7  2 
7
128.6
DETERMINING NUMBER OF SIDES In Exercises 22–25, you are given the
measure of each interior angle of a regular n-gon. Find the value of n.
22. 144o
180  n  2 
n
10
 144
DETERMINING ANGLE MEASURES In Exercises 29–32, you are given the
number of sides of a regular polygon. Find the measure of each exterior angle.
32.
15
360
15
24o
DETERMINING NUMBER OF SIDES In Exercises 33–36, you are given
the measure of each exterior angle of a regular n-gon. Find the value of n.
34.
20o
360
 20
n
18
R
T
U
360
mRTU  mRUT 
 45
8
V
S
mTRU  180   45  45
mTRU  90
W
Z
Y
X