EXAMPLE 1 Find the sum of angle measures in a polygon Find the sum of the measures of the interior angles of a convex octagon. SOLUTION An octagon has 8 sides. Use the Polygon Interior Angles Theorem. (n – 2) 180° = Substitute 8 for n. (8 – 2) 180° Subtract. = 6 180° Multiply. = 1080° ANSWER The sum of the measures of the interior angles of an octagon is 1080°.

Find the number of sides of a polygon The sum of the measures of the interior angles of a convex polygon is 900°. Classify the polygon by the number of sides. SOLUTION Use the Polygon Interior Angles Theorem to write an equation involving the number of sides n. Then solve the equation to find the number of sides. Polygon Interior Angles Theorem (n –2) 180° = 900° Divide each side by 180°. n –2 =5 Add 2 to each side. n = 7 The polygon has 7 sides. It is a heptagon. ANSWER EXAMPLE 2

Find the variable based on the exterior angles SOLUTION Use the Polygon Exterior Angles Theorem to write and solve an equation. Polygon Exterior Angles Theorem x° + 2x° + 89° + 67° = 360° Combine like terms. 3x = 360 Solve for x. x = 68 The correct answer is B. ANSWER EXAMPLE 3

Find angle measures in regular polygons TRAMPOLINE The trampoline shown is shaped like a regular dodecagon. Find (a) the measure of each interior angle and (b) the measure of each exterior angle. SOLUTION a. Use the Polygon Interior Angles Theorem to find the sum of the measures of the interior angles. (n –2) 180° =(12 – 2) 180° = 1800° EXAMPLE 4 Since a dodecagon has 12 congruent interior angles: 1800° 12 = 150 ° Divide

Find angle measures in regular polygons TRAMPOLINE The trampoline shown is shaped like a regular dodecagon. Find (a) the measure of each interior angle and (b) the measure of each exterior angle. SOLUTION b. By the Polygon Exterior Angles Theorem, the sum of the measures of the exterior angles, one angle at each vertex, is 360°. EXAMPLE 4 = 30 ° 360° A dodecagon has 12 exterior angles Divide 12