1. Polygon Angle-Sum Theorems
Skill 30

2. Objective
HSG-SRT.5: Students are responsible for finding the sum of interior and exterior angles of a polygon.

3. Definitions
An equilateral polygon is a polygon with all sides congruent.
An equiangular polygon is a polygon with all angles congruent.
A regular polygon is a polygon that is both equilateral and equiangular.

4. Theorem 37: Polygon Angle-Sum Theorem
The sum of the measures of the interior angles of an n-gon is 180 𝑛 – 2 .
Theorem 38: Corollary to Polygon Angle-Sum Thm.
The measure of each interior angle of a regular n-gon is 180 𝑛−2 𝑛 .
Theorem 39: Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles, one at each vertex, is 360.

5. Example 1; Finding a Polygon Angle-Sum
a) What is the sum of the interior angles of a heptagon.
Heptagon is 7 sides
𝑺𝒖𝒎=𝟏𝟖𝟎 𝒏−𝟐
=𝟏𝟖𝟎 𝟕−𝟐
=𝟏𝟖𝟎 𝟓
=𝟗𝟎𝟎
The sum of the interior angle measures of a heptagon is 900.

6. Example 1; Finding a Polygon Angle-Sum
b) What is the sum of the interior angles of a 17-gon.
17-gon has 17 sides
𝑺𝒖𝒎=𝟏𝟖𝟎 𝒏−𝟐
=𝟏𝟖𝟎 𝟏𝟕−𝟐
=𝟏𝟖𝟎 𝟏𝟓
=𝟐𝟕𝟎𝟎
The sum of the interior angle measures of a 17-gon is 2700.

7. Example 1; Finding a Polygon Angle-Sum
c) The sum of the interior angle measures of a polygon is How many sides does the polygon have?
𝑺𝒖𝒎=𝟏𝟖𝟎 𝒏−𝟐
𝟏𝟗𝟖𝟎=𝟏𝟖𝟎 𝒏−𝟐
𝟏𝟗𝟖𝟎=𝟏𝟖𝟎𝒏−𝟑𝟔𝟎
𝟐𝟑𝟒𝟎=𝟏𝟖𝟎𝒏
𝟏𝟑=𝒏
The polygon is a 13-gon.

8. Example 2; Using the Polygon Angle-Sum Theorem
a) The common housefly has eyes that consist of 4000 facets. Each facet is a regular hexagon. What is the measure of each interior angle in one hexagonal facet?
𝑴𝒆𝒂𝒔𝒖𝒓𝒆 𝒐𝒇 𝒂𝒏 𝑨𝒏𝒈𝒍𝒆= 𝟏𝟖𝟎 𝒏−𝟐 𝒏
= 𝟏𝟖𝟎 𝟔−𝟐 𝟔
= 𝟏𝟖𝟎 𝟒 𝟔
=𝟏𝟐𝟎
The measure of each interior angle in one hexagonal facet is 120.

9. Example 2; Using the Polygon Angle-Sum Theorem
b) What is the measure of each interior angle in a regular nonagon?
𝑴𝒆𝒂𝒔𝒖𝒓𝒆 𝒐𝒇 𝒂𝒏 𝑨𝒏𝒈𝒍𝒆= 𝟏𝟖𝟎 𝒏−𝟐 𝒏
= 𝟏𝟖𝟎 𝟗−𝟐 𝟗
= 𝟏𝟖𝟎 𝟕 𝟗
=𝟏𝟒𝟎
The measure of each interior angle in a nonagon is 140.

10. Example 3; Using Polygon Angle-Sum Theorem
a) What is 𝑚∠𝑌 in pentagon TODAY?
The figure is a pentagon so 𝒏=𝟓
𝒎∠𝑻+𝒎∠𝑶+𝒎∠𝑫+𝒎∠𝑨+𝒎∠𝒀=𝟏𝟖𝟎 𝟓−𝟐
𝟏𝟏𝟎+𝟗𝟎+𝟏𝟐𝟎+𝟏𝟓𝟎+𝒎∠𝒀=𝟏𝟖𝟎 𝟑 T O D A Y
110ᵒ
120ᵒ
150ᵒ
𝟒𝟕𝟎+𝒎∠𝒀=𝟓𝟒𝟎
𝒎∠𝒀=𝟕𝟎

11. Example 3; Using Polygon Angle-Sum Theorem
b) What is 𝑚∠𝐺 in quadrilateral EFGH?
The figure is a quadrilateral so 𝒏=𝟒
𝒎∠𝑬+𝒎∠𝑭+𝒎∠𝑮+𝒎∠𝑯=𝟏𝟖𝟎 𝟒−𝟐
𝟖𝟓+𝟏𝟐𝟎+𝒎∠𝑮+𝟓𝟑=𝟏𝟖𝟎 𝟐
𝟐𝟓𝟖+𝒎∠𝑮=𝟑𝟔𝟎 F E G H
𝒎∠𝑮=𝟏𝟎𝟐
120ᵒ
85ᵒ
53ᵒ

12. Example 4; Finding an Exterior Angle Measure
a) What is the measure of an exterior angle of a regular octagon?
The figure is an octagon so 8 sides
𝟖 𝒎∠𝟏 =𝟑𝟔𝟎
𝒎∠𝟏= 𝟑𝟔𝟎 𝟖
𝒎∠𝟏=𝟒𝟓

13. Example 4; Finding an Exterior Angle Measure
b) What is the measure of an exterior angle of a regular nonagon?
The figure is an nonagon so 9 sides
𝟗 𝒎∠𝟏 =𝟑𝟔𝟎
𝒎∠𝟏= 𝟑𝟔𝟎 𝟗
𝒎∠𝟏=𝟒𝟎

14. #30: Polygon Angle-Sum Theorems
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