5.7 Angle Measures in Polygons

Vocabulary/Theorems  Diagonal: joins 2 nonconsecutive vertices  Convex Polygon: has no vertex going into the interior of the polygon  Concave Polygon: has a vertex going into the interior of the polygon

 Sum of the measures of the interior angles of a convex polygon is 180(n-2) o  Sum of the measures to the exterior angles of a convex polygon is 360 o  In a regular polygon, the interior angles are congruent.

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°.

EXAMPLE 2 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

GUIDED PRACTICE for Examples 1 and 2 The coin shown is in the shape of a regular 11- gon. Find the sum of the measures of the interior angles. 1. ANSWER 1620° The sum of the measures of the interior angles of a convex polygon is 1440°. Classify the polygon by the number of sides. 2. ANSWER decagon

With a partner, do #1-2 on p. 299

EXAMPLE 3 Find an unknown interior angle measure SOLUTION The polygon is a quadrilateral. Write an equation involving x. Then solve the equation. Corollary to Theorem 8.1 x° + 108° + 121° + 59° = 360° Combine like terms. Subtract 288 from each side. x = 72 ALGEBRA Find the value of x in the diagram shown. x = 360 The value of x is 72. ANSWER

GUIDED PRACTICE for Example 3 Use the diagram at the right. Find m S and m T °, 103° ANSWER The measures of three of the interior angles of a quadrilateral are 89°, 110°, and 46°. Find the measure of the fourth interior angle ° ANSWER

EXAMPLE 4 Standardized Test Practice SOLUTION 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

GUIDED PRACTICE for Example 4 A convex hexagon has exterior angles with measures 34°, 49°, 58°, 67°, and 75°. What is the measure of an exterior angle at the sixth vertex? 5. ANSWER 77°

Do #3, 4 on p. 299