Download 8.2 Angles in Polygons

8.2 Angles in Polygons
Goal
Find the measures of interior and exterior angles of polygons.
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 .
2
3
1
4
6 5
n6
Symbols ma1 ma2 … man (n 2) p 180
Follow-Up
The Triangle Sum Theorem states that the sum of the measures
of the angles of a triangle is 180 . To use Theorem 8.1 with a
triangle, substitute 3 for n.
(n 2) p 180 ( 3 2) p 180 1 p 180 180 ✓
The Quadrilateral Interior Angles Theorem states that the sum of
the measures of the interior angles of a quadrilateral is 360 .
To use Theorem 8.1 with a quadrilateral, substitute 4 for n.
(n 2) p 180 ( 4 2) p 180 2 p 180 360 ✓
Example 1
Use the Polygon Interior Angles Theorem
Find the sum of the measures of the interior angles
of the convex heptagon shown.
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 5 p 180 900 Answer The sum of the measures of the interior angles of the
convex heptagon is 900 .
Lesson 8.2 • Geometry, Concepts and Skills Notetaking Guide
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Follow-Up
In Example 1, is your answer true for any convex heptagon?
Yes
Example 2
Find the Measure of an Interior Angle
Find the measure of aA in the diagram.
D
88
Solution
The polygon has 6 sides. Use the Polygon
Interior Angles Theorem and substitute 6 for n.
E
142
C 136
136
B
105 F
A
(n 2) p 180 ( 6 2) p 180 4 p 180 720 Use your answer to write an equation for the sum of the measures
of the interior angles. Solve for maA.
maA maB maC maD maE maF 720 maA 136 136 88 142 105 720 maA 607 720 maA 113 Example 3
Interior Angles of a Regular Polygon
Find the measure of an interior angle of a
regular octagon.
Solution
An octagon has 8 sides. Use the Polygon
Interior Angles Theorem and substitute 8 for n.
(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 to find the measure of one interior angle.
1080 8 135 Answer The measure of an interior angle of a regular octagon
is 135 .
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Geometry, Concepts and Skills Notetaking Guide • Chapter 8
Follow-Up Complete the table for a regular n-gon.
n
3
Sum of the
measures of
the interior
angles
Measure of
one interior
angle
4
5
6
7
8
9
180
360 540 720 900 1080 1260
60
90
108 120 129
135
140
Checkpoint In Exercises 1–3, find the measure of aG.
1.
F
G
2.
3.
H
111
J
103
133
E 118
130 109
D
H
G
72
66
G
58
K
H
J
140 140
K
F 120150 130
L
M
130
4. Find the measure of an interior angle of a regular polygon with
twelve sides.
150
THEOREM 8.2: POLYGON EXTERIOR ANGLES THEOREM
Words The sum of the measures of the exterior
angles of a convex polygon, one angle at each
vertex, is 360 .
2
Symbols ma1 ma2 … man 360 1
3
4
5
n5
Lesson 8.2 • Geometry, Concepts and Skills Notetaking Guide
185
Example 4
Find the Measure of an Exterior Angle
Find the value of x.
85
2x 95
x
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 Polygon Exterior Angles Theorem
3x 180 360
Combine like terms.
3x 180
Subtract 180 from each side.
x 60
Divide each side by 3 .
Answer The value of x is 60 .
Follow-Up Check your answer for Example 4.
95 85 x 2x 95 85 60 2( 60 ) 360 ✓
Checkpoint Find the value of x.
5.
6.
89
3x 60
73
46
x
91
186
Geometry, Concepts and Skills Notetaking Guide • Chapter 8
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21
116