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Exterior Angles CK12 Editor 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-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. Copyright © 2012 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/NonCommercial/Share Alike 3.0 Unported (CC BY-NC-SA) License (http://creativecommons.org/licenses/by-nc-sa/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/terms. Printed: January 22, 2013 AUTHOR CK12 Editor www.ck12.org C ONCEPT Concept 1. Exterior Angles 1 Exterior Angles Learning Objectives • Identify the exterior angles of convex polygons. • Find the sums of exterior angles in convex polygons. Introduction This lesson focuses on the exterior angles in a polygon. There is a surprising feature of the sum of the exterior angles in a polygon that will help you solve problems about regular polygons. Exterior Angles in Convex Polygons Recall that interior means inside and that exterior means outside. So, an exterior angle is an angle on the outside of a polygon. An exterior angle is formed by extending a side of the polygon. As you can tell, there are two possible exterior angles for any given vertex on a polygon. In the figure above we only showed one set of exterior angles; the other set would be formed by extending each side in the opposite (clockwise) direction. However, it doesn’t matter which exterior angles you use because on each vertex their measurement will be the same. Let’s look closely at one vertex, and draw both of the exterior angles that are possible. 1 www.ck12.org As you can see, the two exterior angles at the same vertex are vertical angles. Since vertical angles are congruent, the two exterior angles possible around a single vertex are congruent. Additionally, because the exterior angle will be a linear pair with its adjacent interior angle, it will always be supplementary to that interior angle. As a reminder, supplementary angles have a sum of 180◦ . Example 1 What is the measure of the exterior angle 6 OKL in the diagram below? The interior angle is labeled as 45◦ . Since you need to find the exterior angle, notice that the interior angle and the exterior angle form a linear pair. Therefore the two angles are supplementary—they sum to 180◦ . So, to find the measure of the exterior angle, subtract 45◦ from 180◦ . 180 − 45 = 135 The measure of 6 OKL is 135◦ . Summing Exterior Angles in Convex Polygons By now you might expect that if you add up various angles in polygons, there will be some sort of pattern or rule. For example, you know that the sum of the interior angles of a triangle will always be 180◦ . From that fact, you have learned that you can find the sums of the interior angles of any polygons with n sides using the expression 180(n − 2). There is also a rule for exterior angles in a polygon. Let’s begin by looking at a triangle. To find the exterior angles at each vertex, extend the segments and find angles supplementary to the interior angles. 2 www.ck12.org Concept 1. Exterior Angles The sum of these three exterior angles is: 150◦ + 120◦ + 90◦ = 360◦ So, the exterior angles in this triangle will sum to 360◦ . To compare, examine the exterior angles of a rectangle. In a rectangle, each interior angle measures 90◦ . Since exterior angles are supplementary to interior angles, all exterior angles in a rectangle will also measure 90◦ . 3 www.ck12.org Find the sum of the four exterior angles in a rectangle. 90◦ + 90◦ + 90◦ + 90◦ = 360◦ So, the sum of the exterior angles in a rectangle is also 360◦ . In fact, the sum of the exterior angles in any convex polygon will always be 360◦ . It doesn’t matter how many sides the polygon has, the sum will always be 360◦ . We can prove this using algebra as well as the facts that at any vertex the sum of the interior and one of the exterior angles is always 180◦ , and the sum of all interior angles in a polygon is 180(n − 2). Exterior Angle Sum: The sum of the exterior angles of any convex polygon is 360◦ Proof. At any vertex of a polygon the exterior angle and the interior angle sum to 180◦ . So summing all of the exterior angles and interior angles gives a total of 180 degrees times the number of vertices: (Sum of Exterior Angles) + (Sum of Interior Angles) = 180◦ n. On the other hand, we already saw that the sum of the interior angles was: (Sum of Interior Angles) = 180(n − 2) = 180◦ n − 360◦ . Putting these together we have 180n = (Sum of Exterior Angles) + (Sum of Interior Angles) = (180n − 360) + (Sum of Exterior Angles) 360 = (Sum of Exterior Angles) Example 2 What is m6 ZXQ in the diagram below? 4 www.ck12.org Concept 1. Exterior Angles 6 ZXQ in the diagram is marked as an exterior angle. So, we need to find the measure of one exterior angle on a polygon given the measures of all of the others. We know that the sum of the exterior angles on a polygon must be equal to 360◦ , regardless of how many sides the shape has. So, we can set up an equation where we set all of the exterior angles shown (including m6 ZXQ) summed and equal to 180◦ . Using subtraction, we can find the value of X. 70◦ + 60◦ + 65◦ + 40◦ + m6 ZXQ = 360◦ 235◦ + m6 ZXQ = 360◦ m6 ZXQ = 360◦ − 235◦ m6 ZXQ = 125◦ The measure of the missing exterior angle is 125◦ . We can verify that our answer is reasonable by inspecting the diagram and checking whether the angle in question is acute, right, or obtuse. Since the angle should be obtuse, 125◦ is a reasonable answer (assuming the diagram is accurate). Lesson Summary In this lesson, we explored exterior angles in polygons. Specifically, we have learned: • How to identify the exterior angles of convex polygons. • How to find the sums of exterior angles in convex polygons. We have also shown one example of how knowing the sum of the exterior angles can help you find the measure of particular exterior angles. Review Questions For exercises 1-3, find the measure of each of the labeled angles in the diagram. 1. x = ,y = 5 www.ck12.org 2. w = ,x = 3. a = ,b = ,y = ,z = 4. Draw an equilateral triangle with one set of exterior angles highlighted. What is the measure of each exterior angle? What is the sum of the measures of the three exterior angles in an equilateral triangle? 5. Recall that a regular polygon is a polygon with congruent sides and congruent angles. What is the measure of each interior angle in a regular octagon? 6. How can you use your answer to 5 to find the measure of each exterior angle in a regular octagon? Draw a sketch to justify your answer. 7. Use your answer to 6 to find the sum of the measures of the exterior angles of an octagon. 8. Complete the following table assuming each polygon is a regular polygon. Note: This is similar to a previous exercise with more columns—you can use your answer to that question to help you with this one. 6 www.ck12.org Concept 1. Exterior Angles TABLE 1.1: Regular Polygon name Number of sides Sum of measures of interior angles Measure of each interior angle Measure of each exterior angle Sum of measures of exterior angles triangle 4 5 6 7 octagon decagon 1, 800◦ n 9. Each exterior angle forms a linear pair with its adjacent internal angle. In a regular polygon, you can use two different formulas to find the measure of each exterior angle. One way is to compute 180◦ −(measure of each interior angle). . . in symbols 180 − 180(n−2) . n Alternatively, you can use the fact that all n exterior angles in an n−gon sum to 360◦ and find the measure of each exterior angle with by dividing the sum by n. Again, in symbols this is 360 n Use algebra to show these two expressions are equivalent. Review Answers 1. 2. 3. 4. x = 52◦ , y = 128◦ w = 70◦ , x = 70◦ , y = 110◦ , z = 90◦ a = 107.5◦ , b = 72.5◦ Below is a sample sketch. Each exterior angle measures 120◦ , the sum of the three exterior angles is 360◦ ◦ 5. Sum of the angles is 180(8 − 2) = 1080◦ . So, each angle measures 1080 8 = 135 7 www.ck12.org 6. Since each exterior angle forms a linear pair with its adjacent interior angle, we can find the measure of each exterior angle with 180 − 135 = 45◦ 7. 45(8) = 360◦ TABLE 1.2: Regular Polygon name Number of sides triangle square pentagon hexagon heptagon octagon decagon dodecagon n−gon 3 4 5 6 7 8 10 12 n Sum of measures of interior angles 180◦ 360◦ 540◦ 720◦ 900◦ 1, 080◦ 1, 440◦ 1, 800◦ 180(n − 2)◦ Measure of each interior angle Measure of each exterior angle 60◦ 90◦ 72◦ 60◦ 128.57◦ 135◦ 144◦ 150◦ 120◦ 90◦ 108◦ 120◦ 51.43◦ 45◦ 36◦ 30◦ 180(n−2) ◦ n 360 ◦ n 9. One possible answer. 180 − 8 180(n − 2) 180n 180(n − 2) = − n n n 180n − 180(n − 2) = n 180n − 180n + 360 = n 360 = n Sum of measures of exterior angles 360◦ 360◦ 360◦ 360◦ 360◦ 360◦ 360◦ 360◦ 360◦