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Exterior Angles in Convex Polygons Bill Zahner Lori Jordan Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). Copyright © 2016 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/about/ terms-of-use. Printed: January 6, 2016 AUTHORS Bill Zahner Lori Jordan www.ck12.org C HAPTER Chapter 1. Exterior Angles in Convex Polygons 1 Exterior Angles in Convex Polygons Here you’ll learn the Exterior Angle Sum Theorem that states that the exterior angles of a polygon always add up to 360◦ . What if you were given a twelve-sided regular polygon? How could you determine the measure of each of its exterior angles? After completing this Concept, you’ll be able to use the Exterior Angle Sum Theorem to solve problems like this one. Watch This MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/136947 CK-12 Foundation: Chapter6ExteriorAnglesinConvexPolygonsA Watch the second half of this video. MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/1267 James Sousa: Angles of Convex Polygons Guidance Recall that an exterior angle is an angle on the outside of a polygon and is formed by extending a side of the polygon. 1 www.ck12.org As you can see, there are two sets of exterior angles for any vertex on a polygon. It does not matter which set you use because one set is just the vertical angles of the other, making the measurement equal. In the picture above, the color-matched angles are vertical angles and congruent. The Exterior Angle Sum Theorem stated that the exterior angles of a triangle add up to 360◦ . Let’s extend this theorem to all polygons. Investigation: Exterior Angle Tear-Up Tools Needed: pencil, paper, colored pencils, scissors 1. Draw a hexagon like the hexagons above. Color in the exterior angles as well. 2. Cut out each exterior angle and label them 1-6. 3. Fit the six angles together by putting their vertices together. What happens? The angles all fit around a point, meaning that the exterior angles of a hexagon add up to 360◦ , just like a triangle. We can say this is true for all polygons. Exterior Angle Sum Theorem: The sum of the exterior angles of any polygon is 360◦ . Proof of the Exterior Angle Sum Theorem: Given: Any n−gon with n sides, n interior angles and n exterior angles. Prove: n exterior angles add up to 360◦ NOTE: The interior angles are x1 , x2 , . . . xn . The exterior angles are y1 , y2 , . . . yn . TABLE 1.1: Statement 1. Any n−gon with n sides, n interior angles and n exterior angles. 2 Reason Given www.ck12.org Chapter 1. Exterior Angles in Convex Polygons TABLE 1.1: (continued) Statement 2. xn◦ and y◦n are a linear pair 3. xn◦ and y◦n are supplementary 4. xn◦ + y◦n = 180◦ 5. (x1◦ + x2◦ + . . . + xn◦ ) + (y◦1 + y◦2 + . . . + y◦n ) = 180◦ n 6. (n − 2)180◦ = (x1◦ + x2◦ + . . . + xn◦ ) 7. 180◦ n = (n − 2)180◦ + (y◦1 + y◦2 + . . . + y◦n ) 8. 180◦ n = 180◦ n − 360◦ + (y◦1 + y◦2 + . . . + y◦n ) 9. 360◦ = (y◦1 + y◦2 + . . . + y◦n ) Reason Definition of a linear pair Linear Pair Postulate Definition of supplementary angles Sum of all interior and exterior angles in an n−gon Polygon Sum Formula Substitution PoE Distributive PoE Subtraction PoE Example A What is y? y is an exterior angle, as well as all the other given angle measures. Exterior angles add up to 360◦ , so set up an equation. 70◦ + 60◦ + 65◦ + 40◦ + y = 360◦ y = 125◦ Example B What is the measure of each exterior angle of a regular heptagon? Because the polygon is regular, each interior angle is equal. This also means that all the exterior angles are equal. ◦ ◦ The exterior angles add up to 360◦ , so each angle is 360 7 ≈ 51.43 . Example C What is the sum of the exterior angles in a regular 15-gon? The sum of the exterior angles in any convex polygon, including a regular 15-gon, is 360◦ . Watch this video for help with the Examples above. 3 www.ck12.org MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/136948 CK-12 Foundation: Chapter6ExteriorAnglesinConvexPolygonsB Concept Problem Revisited The exterior angles of a regular polygon sum to 360◦ . The measure of each exterior angle in a dodecagon (twelve◦ ◦ sided regular polygon) is 360 12 = 30 . Guided Practice Find the measure of each exterior angle for each regular polygon below: 1. 12-gon 2. 100-gon 3. 36-gon Answers: For each, divide 360◦ by the given number of sides. 1. 30◦ 2. 3.6◦ 3. 10◦ Explore More 1. What is the measure of each exterior angle of a regular decagon? 2. What is the measure of each exterior angle of a regular 30-gon? 3. What is the sum of the exterior angles of a regular 27-gon? Find the measure of the missing variables: 4. 4 www.ck12.org Chapter 1. Exterior Angles in Convex Polygons 5. 6. The exterior angles of a quadrilateral are x◦ , 2x◦ , 3x◦ , and 4x◦ . What is x? Find the measure of each exterior angle for each regular polygon below: 7. 8. 9. 10. 11. 12. 13. 14. octagon nonagon triangle pentagon 50-gon heptagon 34-gon Challenge Each interior angle forms a linear pair with an exterior angle. In a regular polygon you can use two ◦ (n−2)180◦ ◦ different formulas to find the measure of each exterior angle. One way is 360 n and the other is 180 − n (180◦ minus Equiangular Polygon Formula). Use algebra to show these two expressions are equivalent. 15. Angle Puzzle Find the measures of the lettered angles below given that m || n. Answers for Explore More Problems To view the Explore More answers, open this PDF file and look for section 6.2. 5