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Third Angle Theorem 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: May 8, 2016 AUTHORS Bill Zahner Lori Jordan www.ck12.org C HAPTER Chapter 1. Third Angle Theorem 1 Third Angle Theorem Here you’ll learn the Third Angle Theorem and how to use it to determine information about two triangles with two pairs of angles that are congruent. What if you were given 4FGH and 4XY Z and you were told that 6 F ∼ = 6 X and 6 G ∼ = 6 Y ? What conclusion could you draw about 6 H and 6 Z? After completing this Concept, you’ll be able to make such a conclusion. Watch This MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/136991 CK-12 Foundation: Chapter4TheThirdAngleTheoremA MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/10331 Guidance Find m6 C and m6 J. The sum of the angles in each triangle is 180◦ by the Triangle Sum Theorem. So, for 4ABC, 35◦ +88◦ +m6 C = 180◦ and m6 C = 57◦ . For 4HIJ, 35◦ + 88◦ + m6 J = 180◦ and m6 J is also 57◦ . Notice that we were given that m6 A = m6 H and m6 B = m6 I and we found out that m6 C = m6 J. This can be generalized into the Third Angle Theorem. Third Angle Theorem: If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also congruent. 1 www.ck12.org In other words, for triangles 4ABC and 4DEF, if 6 A ∼ = 6 D and 6 B ∼ = 6 E, then 6 C ∼ = 6 F. Notice that this theorem does not state that the triangles are congruent. That is because if two sets of angles are congruent, the sides could be different lengths. See the picture below. Example A Determine the measure of the missing angles. From the markings, we know that 6 A ∼ = D and 6 E ∼ = 6 B. Therefore, the Third Angle Theorem tells us that 6 C ∼ = 6 F. So, m6 A + m6 B + m6 C = 180◦ m6 D + m6 B + m6 C = 180◦ 42◦ + 83◦ + m6 C = 180◦ m6 C = 55◦ = m6 F Example B The Third Angle Theorem states that if two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also congruent. What additional information would you need to know in order to be able to determine that the triangles are congruent? In order for the triangles to be congruent, you need some information about the sides. If you know two pairs of angles are congruent and at least one pair of corresponding sides are congruent, then the triangles will be congruent. Example C Determine the measure of all the angles in the triangle: 2 www.ck12.org Chapter 1. Third Angle Theorem First we can see that m6 DCA = 15◦ . This means that m6 BAC = 15◦ also because they are alternate interior angles. m6 ABC = 153◦ was given. This means by the Triangle Sum Theorem that m6 BCA = 12◦ . This means that m6 CAD = 12◦ also because they are alternate interior angles. Finally, m6 ADC = 153◦ by the Triangle Sum Theorem. Watch this video for help with the Examples above. MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/136992 CK-12 Foundation: Chapter4TheThirdAngleTheoremB Concept Problem Revisited For two given triangles 4FGH and 4XY Z, you were told that 6 F ∼ = 6 X and 6 G ∼ = 6 Y. By the Third Angle Theorem, 6 H ∼ = 6 Z. Guided Practice Determine the measure of all the angles in the each triangle. 1. 2. 3 www.ck12.org 3. Answers: 1. m6 A = 86, m6 C = 42 and by the Triangle Sum Theorem m6 B = 52. m6 Y = 42, m6 X = 86 and by the Triangle Sum Theorem, m6 Z = 52. 2. m6 C = m6 A = m6 Y = m6 Z = 35. By the Triangle Sum Theorem m6 B = m6 X = 110. 3. m6 A = 28, m6 ABE = 90 and by the Triangle Sum Theorem, m6 E = 62. m6 D = m6 E = 62 because they are alternate interior angles and the lines are parallel. m6 C = m6 A = 28 because they are alternate interior angles and the lines are parallel. m6 DBC = m6 ABE = 90 because they are vertical angles. Explore More Determine the measures of the unknown angles. 1. 2. 3. 4. 4 6 6 6 6 XY Z ZXY LNM MLN www.ck12.org 5. 6. 7. 6 8. 9. 10. 11. 6 Chapter 1. Third Angle Theorem CED GFH FHG 6 6 6 6 6 ACB HIJ HJI IHJ 5 www.ck12.org 12. 13. 14. 15. 6 6 6 6 RQS SRQ T SU TUS Answers for Explore More Problems To view the Explore More answers, open this PDF file and look for section 4.5. 6