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WARM UP

WARM UP. Find the measure of the supplement of a 92° angle. Evaluate (n – 2)180 if n = 12 Solve = 60. SOLUTIONS. 180° - 92° = 88° (12 – 2)180 = 1800 = 60 Multiply both sides by n 60 n = 360 Divide both sides by 60 n = 6. 5.7. Find Angle Measures in Polygons.

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WARM UP

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

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

  3. 5.7 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. ANSWER The sum of the measures of the interior angles of an octagon is 1080°. 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° = 6 180° Subtract. = 1080° Multiply.

  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 = 7 Divide each side by 180°. n = 9 Add 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. 360° 3x + 156 = 68 x = Example 4: Find unknown exterior angle measures SOLUTION Use the Polygon Exterior Angles Theorem to write and solve an equation. x° + 2x° + 89° + 67° = 360 ° Polygon Exterior Angles Theorem Combine like terms. Solve for x.

  11. Checkpoint • Find the sum of the measures of the interior angles of the convex decagon. • The sum of the measures of the interior angles of a convex polygon is 1620°. Classify the polygon by the number of sides. • 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)

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