12-3: Angles of Polygons Big Ideas Math, Course 2 What is the sum of the interior angles of a 102-sided shape?

One triangle = 180⁰ Two triangles = 180 + 180 360⁰ Four triangles = Conceptualizing Triangles Within Polygons What is the total angle measure? One triangle = 180⁰ Two triangles = 180 + 180 360⁰ Four triangles = 4 • 180 720⁰

Knowing the Academic Language Interior Angle Measures of a Polygon The sum S of the interior angle measures of a polygon with n sides is: S = (n – 2) • 180⁰ Shape Sides Formula Sum Triangle 3 (3 – 2) • 180 180⁰ Rectangle 4 (4 – 2) • 180 360⁰ Pentagon 5 (5 – 2) • 180 540⁰ Octagon 8 (8 – 2) • 180 1080⁰ Decagon 10 (10 – 2) • 180 1440⁰

Example 1 Find the sum of the interior angle measures of the shape. a. A 102 sided figure (or a 102-gon) S = (102 – 2) • 180⁰ S = 100 • 180 S = 18000⁰

Finding an Interior Angle Measure of a Polygon To find an interior angle measure: Step 1: Find the sum of the interior angle measure. Step 2: Write and solve an equation to find missing angle measure. Find the value of x. 140⁰ x⁰ 128⁰ 130⁰ 120⁰ 115⁰ 145⁰ Step 1: S = (7 – 2) • 180⁰ S = 900⁰ Step 2: 140 + 145 + 115 + 120 + 130 + 128 + x = 900 778 + x = 900 x = 122

Example 2 Find the value of x. Step 1: S = (5 – 2) • 180⁰ S = 540⁰ 145⁰ 2x⁰ 110⁰ Step 1: S = (5 – 2) • 180⁰ S = 540⁰ Step 2: 145 + 145 + 2x + 110 + 2x = 540 400 + 4x = 540 4x = 140 x = 35

Regular Polygons and Their Angle Measures Regular polygon – all the sides are congruent and all the interior angles are congruent For example: What is the angle measures of a regular octagon? Step 1: Find the sum of the interior angle measures. S = (8 – 2) • 180⁰ S = 1080⁰ Step 2: Divide the sum by the number of interior angles. 1080 ÷ 8 = 135⁰

Finding Exterior Angle Measures Exterior Angle Measures of a Polygon The sum of the measures of the exterior angles of a polygon is 360⁰. e⁰ a⁰ d⁰ b⁰ a + b + c + d + e = 360 c⁰

Examples

Turn to Page 286 in your RAPJ Practice Turn to Page 286 in your RAPJ and do 3, 5, 6 problems