Sum of the Interior Angle Measures This coin was the first to show the image of a real woman.
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1 Interior Angles of a Polygon Sum of the Interior Angle Measures of a Polygon.4 Learning Goals Key Term In this lesson, you will: Write the formula for the sum of the measures of the interior angles of any polygon. Calculate the sum of the measures of the interior angles of any polygon, given the number of sides. Calculate the number of sides of a polygon, given the sum of the measures of the interior angles. Write a formula for the measure of each interior angle of any regular polygon. Calculate the measure of an interior angle of a regular polygon, given the number of sides. Calculate the number of sides of a regular polygon, given the sum of the measures of the interior angles. interior angle of a polygon The Susan B. Anthony dollar coin was minted from 199 to 1981 and then again in This coin was the first to show the image of a real woman. The coin also had an unusual shape inscribed inside of it a regular undecagon, or hendecagon, which is an 11-sided polygon. The shape was no accident, as you can see from the back of the coin which commemorates the Apollo 11 moon landing. 52
2 Problem 1 Inside Job An interior angle of a polygon faces the inside of a polygon and is formed by consecutive sides of the polygon.? 1. Ms. Lambert asked her class to determine the sum of the interior angle measures of a quadrilateral. Carson drew a quadrilateral and added one diagonal as shown. He concluded that the sum of the measures of the interior angles of a quadrilateral must be equal to 360º. Juno drew a quadrilateral and added two diagonals as shown. She concluded that the sum of the measures of the interior angles of a quadrilateral must be equal to 20º. Who is correct? Who is incorrect? Explain your reasoning. 528 Chapter Properties of Quadrilaterals
3 Problem 2 How Many Triangles? The Triangle Sum Theorem states that the sum of the three interior angles of any triangle is equal to 180º. You can use this information to calculate the sum of the interior angles of other polygons. 1. Calculate the sum of the interior angle measures of a quadrilateral by completing each step. a. Draw a quadrilateral. Draw a diagonal using only one vertex of the quadrilateral. A diagonal is a line segment connecting non-adjacent vertices. b. How many triangles are formed when the diagonal divides the quadrilateral? c. If the sum of the interior angle measures of each triangle is 180º, what is the sum of all the interior angle measures of the triangles formed by the diagonal?.4 Sum of the Interior Angle Measures of a Polygon 529
4 2. Calculate the sum of the interior angle measures of a pentagon by completing each step. a. Draw a pentagon. Draw all possible diagonals using only one vertex of the pentagon. b. How many triangles are formed when the diagonal(s) divide the pentagon? c. If the sum of the interior angle measures of each triangle is 180º, what is the sum of all the interior angle measures of the triangles formed by the diagonal(s)? 3. Calculate the sum of the interior angle measures of a hexagon by completing each step. a. Draw a hexagon. Draw all possible diagonals using one vertex of the hexagon. b. How many triangles are formed when the diagonal(s) divide the hexagon? c. If the sum of the interior angle measures of each triangle is 180º, what is the sum of all the interior angle measures of the triangles formed by the diagonal(s)? 530 Chapter Properties of Quadrilaterals
5 4. Complete the table shown. Number of Sides of the Polygon Number of Diagonals Drawn Number of Triangles Formed Sum of the Measures of the Interior Angles 5. What pattern do you notice between the number of possible diagonals drawn from one vertex of the polygon, and the number of triangles formed by those diagonals? 6. Compare the number of sides of the polygon to the number of possible diagonals drawn from one vertex. What do you notice?. Compare the number of sides of the polygon to the number of triangles formed by drawing all possible diagonals from one vertex. What do you notice? 8. What pattern do you notice about the sum of the interior angle measures of a polygon as the number of sides of each polygon increases by 1? 9. Predict the number of possible diagonals drawn from one vertex and the number of triangles formed for a seven-sided polygon using the table you completed. 10. Predict the sum of all the interior angle measures of a seven-sided polygon using the table your completed..4 Sum of the Interior Angle Measures of a Polygon 531
6 11. Continue the pattern to complete the table. Number of Sides of the Polygon Number of Diagonals Drawn Number of Triangles Formed Sum of the Measures of the Interior Angles 12. When you calculated the number of triangles formed in the 16-sided polygon, did you need to know how many triangles were formed in a 15-sided polygon first? Explain your reasoning. 13. If a polygon has 100 sides, how many triangles are formed by drawing all possible diagonals from one vertex? Explain your reasoning. 14. What is the sum of all the interior angle measures of a 100-sided polygon? Explain your reasoning. 15. If a polygon has n sides, how many triangles are formed by drawing all diagonals from one vertex? Explain your reasoning. 16. What is the sum of all the interior angle measures of an n-sided polygon? Explain your reasoning. 1. Use the formula to calculate the sum of all the interior angle measures of a polygon with 32 sides. 18. If the sum of all the interior angle measures of a polygon is 9540º, how many sides does the polygon have? Explain your reasoning. 532 Chapter Properties of Quadrilaterals
7 Problem 3 Regular Stuff 1. Use the formula developed in Problem 2, Question 16 to calculate the sum of all the interior angle measures of a decagon. 2. Calculate each interior angle measure of a decagon if each interior angle is congruent. How did you calculate your answer? 3. Complete the table. Number of Sides of Regular Polygon Sum of Measures of Interior Angles Measure of Each Interior Angle 4. If a regular polygon has n sides, write a formula to calculate the measure of each interior angle. 5. Use the formula to calculate each interior angle measure of a regular 100-sided polygon. 6. If each interior angle measure of a regular polygon is equal to 150º, determine the number of sides. How did you calculate your answer?.4 Sum of the Interior Angle Measures of a Polygon 533
8 Problem 4 The Angle Inside 1. PENTA is a regular pentagon. Solve for x. P E x A N T 2. The Susan B. Anthony dollar coin minted in 1999 features a regular 11-gon, or hendecagon, inside a circle on both sides of the coin. What is the measure of each interior angle of the regular hendecagon? 534 Chapter Properties of Quadrilaterals
9 3. The high school pep squad is preparing a halftime performance for the next basketball game. Six students will hold banners to form a regular hexagon as shown with the school mascot in the very center. Each of the banners is exactly 4 feet long. a. What angle does each student form with his or her banners? Explain your reasoning. b. What is the distance from each student on the regular hexagon to the school mascot in the center? Show your work and explain your reasoning..4 Sum of the Interior Angle Measures of a Polygon 535
10 4. Yael and Brynn wanted to calculate how much space the halftime show would take up. Yael To determine the height of the hexagon, I first drew AC. Then, I drew BD to create two right triangles. Finally, I used trigonometry to solve I already for the lengths AD and DC. know the distance across the hexagon. A 4 ft B D X 4 ft C Brynn To determine the height of the hexagon, I first drew line segments XY and XZ to create triangle XYZ. Then, I drew XW perpendicular to YZ to create two right triangles. Finally, I used trigonometry to solve for the height. Y W Z 4 ft X 536 Chapter Properties of Quadrilaterals
11 a. Verify that both Yael s method and Brynn s method result in the same height..4 Sum of the Interior Angle Measures of a Polygon 53
12 b. Calculate how much rectangular space the students need for the halftime show. Show your work and explain your reasoning. c. About how much space would the halftime show take up if 8 students formed a regular octagon with 4-foot banners? Show your work. Be prepared to share your methods and solutions. 538 Chapter Properties of Quadrilaterals
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heptagon; not regular; hexagon; not regular; quadrilateral; convex concave regular; convex
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