1. 1. Find the measure of the supplement of a 92° angle. 2. Evaluate (n – 2)180 if n = 12 3. Solve = 60

2. 1. 180° - 92° = 88° 2. (12 – 2)180 = 1800 3. = 60 Multiply both sides by n 60 n = 360 Divide both sides by 60 n = 6

3. Find Angle Measures in Polygons

4. VOCABULARY Regular polygon: a polygon that has all congruent sides and angles. Diagonal: A diagonal of a polygon is a segment that joins two nonconsecutive vertices. Interior angles of a polygon: An angle formed by two sides of a polygon. The original angles are the interior angles of the polygon. Exterior angles of a polygon: An angle created by extending side of the polygon outside the figure forming linear pairs with the interior angles.

5. Theorem 5.16: Polygon Interior Angles Theorem The sum of the measure of the interior angles of a convex n-gon is (n – 2)180°. n = 6 Corollary : Interior Angles of Quadrilateral The sum of the measure of the interior angles of a quadrilateral is 360°.

6. Theorem 5.17: Polygon Exterior Angles Theorem The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360 °.

7. 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°.

8. Example2: Find the number of sides of a polygon The sum of the measures of the interior angles of a convex polygon is 1260°. Classify the polygon by the number of sides. 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. (n – 2)180° = 1260°Polygon Interior Angles Theorem n – 2 = 7Divide each side by 180°. n = 9Add 2 to each side. The polygon has 9 sides. It is a nonagon. SOLUTION

9. Example 3: Find an unknown interior angle measure Find the value of x in the diagram shown. The polygon is a quadrilateral. Use the Corollary to the Polygon Interior Angles Theorem to write an equation involving x. Then solve the equation. x°+ 108° + 121° + 59° = 360° Corollary 8.1 x°+ 288° = 360 °Combine Like Terms. x° = 72°Solve for x. SOLUTION

10. Example 4: Find unknown exterior angle measures 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 + 156 = 360° Solve for x. x = 68

11. Checkpoint 1. Find the sum of the measures of the interior angles of the convex decagon. 2. The sum of the measures of the interior angles of a convex polygon is 1620°. Classify the polygon by the number of sides. 3. Use the diagram at the right. N Find m ‘s K and L. J M K L

12. Checkpoint Cont…. 4. A convex pentagon has exterior angles with measures 66°, 77°, 82°, and 62°. What is the measure of an exterior angle at the fifth vertex? 5. Find the measure of (a) each interior angle and (b) each exterior angle of a regular nonagon.

13. Homework p. 300-301 #’s 2 – 36 (even)