Polygon Angle-Sum Theorems Skill 30

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

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.

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.

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.

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.

Example 1; Finding a Polygon Angle-Sum c) The sum of the interior angle measures of a polygon is 1980. 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.

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.

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.

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

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

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

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

#30: Polygon Angle-Sum Theorems Questions? Summarize Notes Homework Video Quiz