Page 1 of 7
8.2
Goal
Find the measures of
interior and exterior
angles of polygons.
Angles in Polygons
The definitions for interior angles and exterior angles can be extended
to include angles formed in any polygon. In the diagrams shown
below, interior angles are red, and exterior angles are blue.
Key Words
interior angles
• interior angle p. 181
exterior angles
• exterior angle p. 181
Activity 8.2 suggests the Polygon Interior Angles Theorem shown below.
THEOREM 8.1
Polygon Interior Angles Theorem
Words
The sum of the measures of the
interior angles of a convex polygon
with n sides is (n 2) p 180.
Symbols
EXAMPLE
2
3
1
4
6
5
ma1 ma2 . . . man (n 2) p 180
1
n6
Use the Polygon Interior Angles Theorem
Find the sum of the measures of the
interior angles of a convex heptagon.
Solution
A heptagon has 7 sides. Use the Polygon Interior Angles Theorem
and substitute 7 for n.
(n 2) p 180 (7 2) p 180
ANSWER
Substitute 7 for n.
5 p 180
Simplify.
900
Multiply.
The sum of the measures of the interior angles of a
convex heptagon is 900.
8.2
Angles in Polygons
417
Page 2 of 7
EXAMPLE
2
Find the Measure of an Interior Angle
Find the measure of aA in the diagram.
D
88
Solution
E
142
105 F
C 136
The polygon has 6 sides, so the sum of the
measures of the interior angles is:
136
B
A
(n 2) p 180 (6 2) p 180 4 p 180 720.
Add the measures of the interior angles and set the sum equal to 720.
136 136 88 142 105 maA 720
Student Help
The sum is 720.
607 maA 720
SKILLS REVIEW
To review solving an
equation, see p. 671.
Simplify.
maA 113
ANSWER
Subtract 607.
The measure of aA is 113.
EXAMPLE
3
Interior Angles of a Regular Polygon
Find the measure of an interior angle of
a regular octagon.
Solution
The sum of the measures of the interior
angles of any octagon is:
(n 2) p 180 (8 2) p 180 6 p 180 1080.
Because the octagon is regular, each angle has the same measure.
So, divide 1080 by 8 to find the measure of one interior angle.
1080
135
8
ANSWER
The measure of an interior angle of a regular octagon is 135.
Find Measures of Interior Angles
In Exercises 1–3, find the measure of aG.
1.
F
111
E 118
130
D
G
2.
103
109
H
3.
H
G
J
G
133
H
J
140 140
120
F
58
150 130
M
L
K
4. Find the measure of an interior angle of a regular polygon with
twelve sides.
418
Chapter 8
Polygons and Area
K
Page 3 of 7
Exterior Angles The diagrams below show that the sum of the
measures of the exterior angles of the convex polygon is 360.
1
●
Visualize It!
A circle contains two
straight angles. So,
there are 180 180,
or 360, in a circle.
180
180
Cut out the
exterior angles.
3
●
Arrange the exterior
angles to form 360.
360
1
2
1
2
1
5
360
or
2
●
Shade one exterior
angle at each vertex.
5
4
2
3
3
5
3
4
4
The sum of the measures of the exterior angles of a convex polygon does
not depend on the number of sides that the polygon has.
THEOREM 8.2
Polygon Exterior Angles Theorem
Words
3
2
The sum of the measures of the
exterior angles of a convex polygon,
one angle at each vertex, is 360.
Symbols
EXAMPLE
4
1
ma1 ma2 . . . man 360
4
5
n5
Find the Measure of an Exterior Angle
Find the value of x.
85
Solution
Using the Polygon Exterior Angles
Theorem, set the sum of the measures
of the exterior angles equal to 360.
95 85 2x x 360
180 3x 360
3x 180
x 60
ANSWER
2x
95
x
Polygon Exterior Angles Theorem
Combine like terms.
Subtract 180 from each side.
Divide each side by 3.
The value of x is 60.
8.2
Angles in Polygons
419
Page 4 of 7
Find Exterior Angle Measures
Find the value of x.
5.
6.
3x
89
60
73
x
46
7.
8.
x
x
116
92
3x
70
2x
110
75
65
8.2 Exercises
Guided Practice
Vocabulary Check
1. Name an interior angle and an
C
A
F
Skill Check
G
B
exterior angle of polygon ABCDE
shown at the right.
E
H
D
2. Write the formula that is used to find the sum of the measures
of the interior angles of any convex polygon with n sides.
3. Use your answer from Exercise 2 to find the sum of the measures
of the interior angles of a convex polygon with 9 sides.
Write an equation to find the measure of a1. Do not solve
the equation.
4.
5.
1
100
130
65
1
40
6.
7.
1
115
420
Chapter 8
Polygons and Area
66
86
57
59
89
117
85
1
Page 5 of 7
Practice and Applications
Extra Practice
See p. 689.
Sum of Interior Angle Measures Find the sum of the measures of
the interior angles of the convex polygon.
8.
9.
10.
Polygons with n Sides Find the sum of the measures of the interior
angles of the convex polygon with n sides.
11. n 10
12. n 15
13. n 20
14. n 30
15. n 52
16. n 100
Interior Angle Measures Find the measure of aA.
17.
18.
B
B
110
A
92
A
149 C
F 100
D
101
68
19.
E
C
20.
A
B
E
113
135
91
C
102 146
B
120 D
124
80 130
D
C
IStudent Help
ICLASSZONE.COM
HOMEWORK HELP
Extra help with problem
solving in Exs. 21–23 is
at classzone.com
158
F
A
D
E
Interior Angles of Regular Polygons Find the measure of an interior
angle of the regular polygon.
21.
22.
23.
Using Algebra Find the value of x.
24.
25.
Homework Help
Example 1:
Example 2:
Example 3:
Example 4:
Exs. 8–16
Exs. 17–20
Exs. 21–23
Exs. 24, 25
75
85
44
65
54
71
44
x
76
2x
8.2
3x
Angles in Polygons
421
Page 6 of 7
Baseball A home plate for a baseball field is a pentagon as shown.
26. Is the polygon regular? Explain why or
Sports
A
B
E
C
why not.
27. What is the sum of the interior angles?
28. Find the measures of aC and aE.
D
BASEBALL Home plate is
used to find the placement of
the foul lines on the playing
field.
Visualize It! Cut a strip of lined paper and tie it into an overhand knot
as shown. Gently flatten the knot to form a pentagon. The pentagon
should be regular.
29. Find the measure of an interior angle of a regular pentagon.
30. Measure the interior angles of your knot with a protractor. Use
your answer to Exercise 29 to determine whether your pentagon
is regular.
Logical Reasoning Select the word that makes the statement true.
31. The sum of the measures of the exterior angles of a convex
polygon, one at each vertex, is (always, sometimes, never) 360.
32. The sum of the measures of the interior angles of a convex
octagon is (always, sometimes, never) 1440.
33. The measures of the exterior angles of a convex polygon are
(always, sometimes, never) equal.
34. Outdoor Furniture You are
constructing a regular hexagonal
wooden bench like the one shown.
On the bench, a1 c a2. Find the
measures of a1 and a2 so that
you know what angle to use to cut
the pieces of wood.
1
Challenge Find the number of sides of the regular polygon with the
given exterior angle measure.
35. 60
422
Chapter 8
Polygons and Area
36. 20
37. 72
38. 10
2
Page 7 of 7
Standardized Test
Practice
Multiple Choice In Exercises 39 and 40, use the diagram below.
39. What is the value of x?
A
C
B
D
18
117
R
x
78
P
130
P 2x
198
40. What is the measure of aP?
F
Mixed Review
G
39
U
H
78
134 S
120
T
102
J
156
234
Using a Centroid E is the centroid of T ABC. Find BE and ED.
(Lesson 4.6)
41. BD 6
42. BD 33
43. BD 52
A
A
A
D
C
E
D
D
E
B
C
B
E
C
B
Algebra Skills
Absolute Value Evaluate. (Skills Review, p. 662)
44. 5
45. 11
46. 48
47. 0
48. 13.2
49. 2
50. 0.001
51. 1.11
Quiz 1
Decide whether the polygon is regular. Explain your answer.
(Lesson 8.1)
1.
2.
3.
Find the measure of a1. (Lesson 8.2)
4.
112
5.
1
120
6.
140
100
92
76
1
104
68
140 120
7.
1
79
100
8.
92
61
74
93
95
1
50
1
44
8.2
Angles in Polygons
423