Lesson 6-1 Angles of Polygons TARGETS Find and use the sum of the measures of the interior angles of a polygon. Find and use the sum of the measures of the exterior angles of a polygon. Targets

Polygon interior angles sume LESSON 6.1: Angles of Polygons Polygon Interior Angles Sum # of Sides Angle Measure Sum of Angles 3 mA= mC= mB= 4 mF= mH= mG= mI= 5 mP= mS= mQ= mT= mR= Polygon interior angles sume

Polygon interior angles sume LESSON 6.1: Angles of Polygons Polygon Interior Angles Sum The sum of the interior angle measures of an n-sided convex polygon is 180(n - 2) Polygon interior angles sume

Answer: The sum of the measures is 1260. LESSON 6.1: Angles of Polygons EXAMPLE 1 Find the Interior Angles Sum of a Polygon A. Find the sum of the measures of the interior angles of a convex nonagon. A nonagon has nine sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures. (n – 2) ● 180 = (9 – 2) ● 180 n = 9 = 7 ● 180 or 1260 Simplify. Answer: The sum of the measures is 1260. Example 1A

B. Find the measure of each interior angle of parallelogram RSTU. LESSON 6.1: Angles of Polygons EXAMPLE 1 Find the Interior Angles Sum of a Polygon B. Find the measure of each interior angle of parallelogram RSTU. Step 1 Find x. Since the sum of the measures of the interior angles is Write an equation to express the sum of the measures of the interior angles of the polygon. Example 1B

Sum of measures of interior angles Find the Interior Angles Sum of a Polygon Sum of measures of interior angles Substitution Combine like terms. Subtract 8 from each side. Divide each side by 32. Example 1B

Step 2 Use the value of x to find the measure of each angle. Find the Interior Angles Sum of a Polygon Step 2 Use the value of x to find the measure of each angle. m  R = 5x = 5(11) or 55 m  S = 11x + 4 = 11(11) + 4 or 125 m  T = 5x = 5(11) or 55 m  U = 11x + 4 = 11(11) + 4 or 125 Answer: mR = 55, mS = 125, mT = 55, mU = 125 Example 1B

Find the measure of each interior angle of a regular octagon. LESSON 6.1: Angles of Polygons EXAMPLE 2 Interior Angle Measure of Regular Polygon Find the measure of each interior angle of a regular octagon. Example 2

S = 180(n – 2) Interior Angle Sum Theorem (150)n = 180(n – 2) S = 150n LESSON 6.1: Angles of Polygons EXAMPLE 3 Find Number of Sides Given Interior Angle Measure The measure of an interior angle of a regular polygon is 150. Find the number of sides in 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 (150)n = 180(n – 2) S = 150n 150n = 180n – 360 Distributive Property 0 = 30n – 360 Subtract 150n from each side. Example 3

Answer: The polygon has 12 sides. Find Number of Sides Given Interior Angle Measure 360 = 30n Add 360 to each side. 12 = n Divide each side by 30. Answer: The polygon has 12 sides. Example 3

Polygon Exterior Angles Sum LESSON 6.1: Angles of Polygons Polygon Exterior Angles Sum The sum of the exterior angle measures of a convex polygon, one angle at each vertex, is 360 Polygon Ext Angles Sum

A. Find the value of x in the diagram. LESSON 6.1: Angles of Polygons EXAMPLE 4 Find Exterior Angle Measures of a Polygon A. Find the value of x in the diagram. Example 4A

Find Exterior Angle Measures of a Polygon Use the Polygon Exterior Angles Sum Theorem to write an equation. Then solve for x. 5x + (4x – 6) + (5x – 5) + (4x + 3) + (6x – 12) + (2x + 3) + (5x + 5) = 360 (5x + 4x + 5x + 4x + 6x + 2x + 5x) + [(–6) + (–5) + 3 + (–12) + 3 + 5] = 360 31x – 12 = 360 31x = 372 x = 12 Answer: x = 12 Example 4A

B. Find the measure of each exterior angle of a regular decagon. LESSON 6.1: Angles of Polygons EXAMPLE 4 Find Exterior Angle Measures of a Polygon B. Find the measure of each exterior angle of a regular decagon. A regular decagon has 10 congruent sides and 10 congruent angles. The exterior angles are also congruent, since angles supplementary to congruent angles are congruent. Let n = the measure of each exterior angle and write and solve an equation. 10n = 360 Polygon Exterior Angle Sum Theorem n = 36 Divide each side by 10. Answer: The measure of each exterior angle of a regular decagon is 36. Example 4B