Angles of Polygons Example 1 Interior Angles of Regular Polygons ARCHITECTURE The Pentagon in Washington, D.C. is shaped like a regular pentagon. Find the sum of the measures of the interior angles of the largest pentagon-shaped section of the Pentagon building. Since the Pentagon is a convex polygon, we can use the Interior Angle Sum Theorem. S = 180(n - 2) Interior Angle Sum Theorem = 180(5 - 2) n = 5 = 180(3) or 540 Simplify. The sum of the measures of the interior angles is 540. Example 2 Sides of a Polygon The measure of an interior angle of a regular polygon is 135. Find the number of sides of the polygon. Use the Interior Angle Sum Theorem to write an equation to solve for n, the number of sides. S = 180(n - 2) Interior Angle Sum Theorem (135)n = 180(n - 2) S = 135n 135n = 180n - 360 Distributive Property 360 = 45n Subtract 135n and add 360 to each side. 8 = n Divide each side by 45. The polygon has 8 sides. Example 3 Interior Angles of Nonregular Polygons ALGEBRA Find the measure of each interior angle. Since n = 4, the sum of the measures of the interior angles is 180(4 – 2) or 360. Write an equation to express the sum of the measures of the interior angles of the polygon. 360 = m R + m S + m T + m U Sum of measures of angles 360 = x + 2x + 3x + 4x Substitution 360 = 10x Combine like terms. 36 = x Divide each side by 10. Use the value of x to find the measure of each angle. m R = 36, m S = 2 ∙ 36 or 72, m T = 3 ∙ 36 or 108, m U = 4 ∙ 36 or 144. Example 4 Exterior Angles Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHI. At each vertex, extend a side to form one exterior angle. The sum of the measures of the exterior angles is 360. A convex regular nonagon has 9 congruent exterior angles. 9n = 360 n = measure of each exterior angle n = 40 Divide each side by 9. The measure of each exterior angle is 40. Since each exterior angle and its corresponding interior angle form a linear pair, the measure of the interior angle is 180 – 40 or 140. .