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Polygon Interior Angle Measures and Sums

Learn how to find the interior angle measures and sum of a polygon, including real-world examples and applications. Also explore the concept of the sum of exterior angles of a polygon.

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Polygon Interior Angle Measures and Sums

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  1. Splash Screen

  2. Five-Minute Check (over Chapter 5) 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

  3. A B C State whether this sentence is always, sometimes, or never true. The three altitudes of a triangle intersect at a point inside the triangle. A. always B. sometimes C. never 5-Minute Check 1

  4. A B C D A.n = 8; B.n = 8; C.n = 6; D.n = 6; Find n and list the sides of ΔPQR in order from shortest to longest if mP = 12n – 15, mQ = 7n + 26, and mR = 8n – 47. 5-Minute Check 2

  5. A B C D State the assumption you would make to start an indirect proof of the statement. If –2x≥ 18, then x≤ –9. A.x is positive. B. –2x < 18 C.x > –9 D.x < 9 5-Minute Check 3

  6. A B C D Find the range for the measure of the third side of a triangle given that the measures of two sides are 43 and 29. A.n > 14 B. 14 < n < 72 C. 29 < n < 43 D.n > 0 5-Minute Check 4

  7. A B C D Write an inequality relating mABD and mCBD. A.mABD < mCBD B.mABD ≤ mCBD C.mABD > mCBD D.mABD = mCBD 5-Minute Check 5

  8. A B C D 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

  9. You named and classified polygons. (Lesson 1–6) • 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

  10. diagonal Vocabulary

  11. Concept 1

  12. 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

  13. 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. Find the Interior Angles Sum of a Polygon B. Find the measure of each interior angle of parallelogram RSTU. Step 1 Find x. Example 1B

  14. 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

  15. 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

  16. A B C D A. Find the sum of the measures of the interior angles of a convex octagon. A. 900 B. 1080 C. 1260 D. 1440 Example 1A

  17. A B C D B. Find the value of x. A.x = 7.8 B.x = 22.2 C.x = 15 D.x = 10 Example 1B

  18. Interior Angle Measure of Regular Polygon ARCHITECTUREA 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

  19. Interior Angle Measure of Regular Polygon Understand Look at the diagram of the situation. The measure of the angle of a corner in between two walkways is the interior angle of a regular pentagon. Plan Use the Polygon Interior Angles Sum Theorem to find the sum of the measures of the angles. Since the angles of a regular polygon are congruent, divide this sum by the number of angles to find the measure of each interior angle. Example 2

  20. 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

  21. 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

  22. A B C D A pottery mold makes bowls that are in the shape of a regular heptagon. Find the measure of one of the interior angles of the bowl. A. 130° B. 128.57° C. 140° D. 125.5° Example 2

  23. 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

  24. 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

  25. A B C D The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. A. 12 B. 9 C. 11 D. 10 Example 3

  26. Concept 2

  27. Find Exterior Angle Measures of a Polygon A. Find the value of x in the diagram. Example 4A

  28. 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

  29. 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

  30. A B C D A. Find the value of x in the diagram. A. 10 B. 12 C. 14 D. 15 Example 4A

  31. A B C D B. Find the measure of each exterior angle of a regular pentagon. A. 72 B. 60 C. 45 D. 90 Example 4B

  32. End of the Lesson

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