1. 8.2
ESSENTIAL QUESTION:
How do you calculate the measures of interior and exterior angles of polygons?

2. Theorem 8.1 Polygon Interior Angles Theorem
The sum of the measures of the interior angles of a convex polygon with n sides is

4. Example 1
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) · 180° = (7 – 2) · 180°
Substitute 7 for n.
= 5 · 180°
Simplify.
= 900°
Multiply.
The sum of the measures of the interior angles of a convex heptagon is 900°.
ANSWER 4

5. Find the measure of A in the diagram.
Example 2
Find the Measure of an Interior Angle
Find the measure of A in the diagram.
SOLUTION
The polygon has 6 sides, so the sum of the measures of the interior angles is:
(n – 2) · 180° = (6 – 2) · 180° = 4 · 180° = 720°.
Add the measures of the interior angles and set the sum equal to 720°.
136° + 136° + 88° + 142° + 105° + mA = 720°
The sum is 720°.
607° + mA = 720°
Simplify.
mA = 113°
Subtract 607°.
The measure of A is 113°.
ANSWER 5

6. Find the measure of an interior angle of a regular octagon.
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) · 180° = (8 – 2) · 180° = 6 · 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. 8
1080°
= 135°
The measure of an interior angle of a regular octagon is 135°.
ANSWER 6

7. In Exercises 1–3, find the measure of G.
Checkpoint
Find Measures of Interior Angles
In Exercises 1–3, find the measure of G. 2. 1. 3.
Find the measure of an interior angle of a regular polygon with twelve sides. 4.

8. In Exercises 1–3, find the measure of G. 1. ANSWER 72°
Checkpoint
Find Measures of Interior Angles
In Exercises 1–3, find the measure of G. 1.
ANSWER
72° 2.
ANSWER
66° 3.
ANSWER
130°
Find the measure of an interior angle of a regular polygon with twelve sides. 4.
ANSWER
150°

11. Theorem 8.2 Polygon Exterior Angles Theorem
The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360°.

13. Using the Polygon Exterior Angles Theorem, set the sum of the measures
Example 4
Find the Measure of an Exterior Angle
Find the value of x.
Using the Polygon Exterior Angles
Theorem, set the sum of the measures
of the exterior angles equal to 360°.
SOLUTION
95° + 85° + 2x° + x° = 360°
Polygon Exterior Angles Theorem
x = 360
Combine like terms.
Subtract 180 from each side.
3x = 180
Divide each side by 3.
x = 60
The value of x is 60.
ANSWER 13

14. Checkpoint Find the value of x. 5. 6. ANSWER 91 ANSWER 21 7. 8. ANSWER
Find Exterior Angle Measures
Find the value of x. 5. 6.
ANSWER 91
ANSWER 21 7. 8.
ANSWER 30
ANSWER
125

15. Worksheet 8.2B
Quiz Tomorrow Section 8.1 & 8.2