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Alternate Interior Angles 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 4, 2016 AUTHORS Bill Zahner Lori Jordan www.ck12.org C HAPTER Chapter 1. Alternate Interior Angles 1 Alternate Interior Angles Here you’ll learn what alternate interior angles are and their relationship with parallel lines. What if you were presented with two angles that are on opposite sides of a transversal, but inside the lines? How would you describe these angles and what could you conclude about their measures? After completing this Concept, you’ll be able to answer these questions using your knowledge of alternate interior angles. Watch This MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/136883 CK-12 Foundation: Chapter3AlternateInteriorAnglesA Watch the portions of this video dealing with alternate interior angles. MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/1328 James Sousa: Angles and Transversals MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/1339 James Sousa: Proof that Alternate Interior Angles Are Congruent MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/1340 James Sousa: Proof of Alternate Interior Angles Converse 1 www.ck12.org Guidance Alternate Interior Angles are two angles that are on the interior of l and m, but on opposite sides of the transversal. 6 3 and 6 6 are alternate interior angles. Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Proof of Alternate Interior Angles Theorem: Given: l || m Prove: 6 3 ∼ =6 6 TABLE 1.1: Statement 1. l || m 2. 6 3 ∼ =6 7 6 3. 7 ∼ =6 6 4. 6 3 ∼ =6 6 Reason Given Corresponding Angles Postulate Vertical Angles Theorem Transitive PoC There are several ways we could have done this proof. For example, Step 2 could have been 6 2 ∼ = 6 6 for the same ∼ ∼ 6 6 6 6 reason, followed by 2 = 3. We could have also proved that 4 = 5. Converse of Alternate Interior Angles Theorem: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. Example A Find m6 1. 2 www.ck12.org Chapter 1. Alternate Interior Angles m6 2 = 115◦ because they are corresponding angles and the lines are parallel. m6 1 = 115◦ also. 6 6 1 and 6 2 are vertical angles, so 1 and the 115◦ angle are alternate interior angles. Example B Find the measure of the angle and x. The two given angles are alternate interior angles so, they are equal. Set the two expressions equal to each other and solve for x. (4x − 10)◦ = 58◦ 4x = 68◦ x = 17◦ Example C Prove the Converse of the Alternate Interior Angles Theorem. Given: l and m and transversal t 6 3∼ =6 6 Prove: l || m TABLE 1.2: Statement 1. l and m and transversal t 6 3 ∼ =6 6 Reason Given 3 www.ck12.org TABLE 1.2: (continued) Statement 2. 6 3 ∼ =6 2 6 3. 2 ∼ =6 6 4. l || m Reason Vertical Angles Theorem Transitive PoC Converse of the Corresponding Angles Postulate 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/136884 CK-12 Foundation: Chapter3AlternateInteriorAnglesB Guided Practice 1. Is l || m? 2. What does x have to be to make a || b? 3. List the pairs of alternate interior angles: 4 www.ck12.org Chapter 1. Alternate Interior Angles Answers: 1. First, find m6 1. We know its linear pair is 109◦ . By the Linear Pair Postulate, these two angles add up to 180◦ , so m6 1 = 180◦ − 109◦ = 71◦ . This means that l || m, by the Converse of the Corresponding Angles Postulate. 2. Because these are alternate interior angles, they must be equal for a || b. Set the expressions equal to each other and solve. 3x + 16◦ = 5x − 54◦ 70◦ = 2x 35◦ = x To make a || b, x = 35◦ . 3. Alternate Interior Angles: 6 4 and 6 5, 6 3 and 6 6. Interactive Practice MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/113006 Explore More 1. Is the angle pair 6 6 and 6 3 congruent, supplementary or neither? 2. Give two examples of alternate interior angles in the diagram: 5 www.ck12.org For 3-4, find the values of x. 3. 4. For question 5, use the picture below. Find the value of x. 5. m6 4 = (5x − 33)◦ , m6 5 = (2x + 60)◦ 6. Are lines l and m parallel? If yes, how do you know? For 7-12, what does the value of x have to be to make the lines parallel? 6 www.ck12.org 7. 8. 9. 10. 11. 12. m6 m6 m6 m6 m6 m6 Chapter 1. Alternate Interior Angles 4 = (3x − 7)◦ and m6 5 = (5x − 21)◦ 3 = (2x − 1)◦ and m6 6 = (4x − 11)◦ 3 = (5x − 2)◦ and m6 6 = (3x)◦ 4 = (x − 7)◦ and m6 5 = (5x − 31)◦ 3 = (8x − 12)◦ and m6 6 = (7x)◦ 4 = (4x − 17)◦ and m6 5 = (5x − 29)◦ For questions 13-15, use the picture below. 13. What is the alternate interior angle to 6 4? 14. What is the alternate interior angle to 6 5? 15. Are the two lines parallel? Explain. Answers for Explore More Problems To view the Explore More answers, open this PDF file and look for section 3.4. 7