Five-Minute Check (over Chapter 5) CCSS Then/Now New Vocabulary Theorem 6.1: Polygon Interior Angles Sum Example 1: Find the Interior Angles Sum of a Polygon Example 2: Real-World Example: Interior Angle Measure of Regular Polygon Example 3: Find Number of Sides Given Interior Angle Measure Theorem 6.2: Polygon Exterior Angles Sum Example 4: Find Exterior Angle Measures of a Polygon Lesson Menu

Write an equation that you can use to find the measures of the angles of the triangle. A. x – 5 + 3x = 180 B. x – 5 + 3x + 111 = 180 C. x – 5 + 3x = 69 D. x – 5 + 3x = 111 5-Minute Check 6

Write an equation that you can use to find the measures of the angles of the triangle. A. x – 5 + 3x = 180 B. x – 5 + 3x + 111 = 180 C. x – 5 + 3x = 69 D. x – 5 + 3x = 111 5-Minute Check 6

Mathematical Practices 4 Model with mathematics. Content Standards G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Mathematical Practices 4 Model with mathematics. 3 Construct viable arguments and critique the reasoning of others. CCSS

You named and classified polygons. 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. Then/Now

diagonal Vocabulary

Concept 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: Example 1A

Answer: The sum of the measures is 1260. 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. 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: 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

A. Find the sum of the measures of the interior angles of a convex octagon. B. 1080 C. 1260 D. 1440 Example 1A

A. Find the sum of the measures of the interior angles of a convex octagon. B. 1080 C. 1260 D. 1440 Example 1A

Interior Angle Measure of Regular Polygon ARCHITECTURE A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the measure of one of the interior angles of the pentagon. Example 2

Solve Find the sum of the interior angle measures. Interior Angle Measure of Regular Polygon Solve Find the sum of the interior angle measures. (n – 2) ● 180 = (5 – 2) ● 180 n = 5 = 3 ● 180 or 540 Simplify. Find the measure of one interior angle. Substitution Divide. Example 2

Interior Angle Measure of Regular Polygon Answer: The measure of one of the interior angles of the food court is 108. Check To verify that this measure is correct, use a ruler and a protractor to draw a regular pentagon using 108 as the measure of each interior angle. The last side drawn should connect with the beginning point of the first segment drawn. Example 2

Concept 2

A. Find the value of x in the diagram. 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: 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. 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: Example 4B

B. Find the measure of each exterior angle of a regular decagon. 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

A. Find the value of x in the diagram. B. 12 C. 14 D. 15 Example 4A

A. Find the value of x in the diagram. B. 12 C. 14 D. 15 Example 4A

B. Find the measure of each exterior angle of a regular pentagon. Example 4B

B. Find the measure of each exterior angle of a regular pentagon. Example 4B