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AA Similarity 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: February 26, 2016 AUTHORS Bill Zahner Lori Jordan www.ck12.org C HAPTER Chapter 1. AA Similarity 1 AA Similarity Here you’ll learn how to determine whether or not two triangles are similar using AA Similarity. What if you were given a pair of triangles and the angle measures for two of their angles? How could you use this information to determine if the two triangles are similar? After completing this Concept, you’ll be able to use AA Similarity to decide if two triangles are similar. Watch This MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/137606 CK-12 Foundation: Chapter7AASimilarityA Watch this video beginning at the 2:09 mark. MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/1302 James Sousa: Similar Triangles MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/1357 James Sousa: Similar Triangles by AA Guidance The Third Angle Theorem states if two angles are congruent to two angles in another triangle, the third angles are congruent too. Because a triangle has 180◦ , the third angle in any triangle is 180◦ minus the other two angle measures. Let’s investigate what happens when two different triangles have the same angle measures. 1 www.ck12.org Investigation: Constructing Similar Triangles Tools Needed: pencil, paper, protractor, ruler 1. Draw a 45◦ angle. Extend the horizontal side and then draw a 60◦ angle on the other side of this side. Extend the other side of the 45◦ angle and the 60◦ angle so that they intersect to form a triangle. What is the measure of the third angle? Measure the length of each side. 2. Repeat Step 1 and make the horizontal side between the 45◦ and 60◦ angle at least 1 inch longer than in Step 1. This will make the entire triangle larger. Find the measure of the third angle and measure the length of each side. 3. Find the ratio of the sides. Put the sides opposite the 45◦ angles over each other, the sides opposite the 60◦ angles over each other, and the sides opposite the third angles over each other. What happens? AA Similarity Postulate: If two angles in one triangle are congruent to two angles in another triangle, the two triangles are similar. The AA Similarity Postulate is a shortcut for showing that two triangles are similar. If you know that two angles in one triangle are congruent to two angles in another, which is now enough information to show that the two triangles are similar. Then, you can use the similarity to find the lengths of the sides. Example A Determine if the following two triangles are similar. If so, write the similarity statement. Find the measure of the third angle in each triangle. m6 G = 48◦ and m6 M = 30◦ by the Triangle Sum Theorem. Therefore, all three angles are congruent, so the two triangles are similar. 4FEG ∼ 4MLN. 2 www.ck12.org Chapter 1. AA Similarity Example B Determine if the following two triangles are similar. If so, write the similarity statement. m6 C = 39◦ and m6 F = 59◦ . The angles are not equal, 4ABC and 4DEF are not similar. Example C Are the following triangles similar? If so, write the similarity statement. Because AE || CD, 6 A ∼ = 6 D and 6 C ∼ = 6 E by the Alternate Interior Angles Theorem. Therefore, by the AA Similarity Postulate, 4ABE ∼ 4DBC. Watch this video for help with the Examples above. MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/137607 CK-12 Foundation: Chapter7AASimilarityB Guided Practice Are the following triangles similar? If so, write a similarity statement. 1. 3 www.ck12.org 2. 3. Answers: 1. Yes, 4DGE ∼ 4FGD ∼ 4FDE. 2. Yes, 4HLI ∼ 4HMJ. 3. No, though 6 MNQ ∼ = 6 ONP because they are vertical angles, we need to have two pairs of congruent angles in order to be able to say that the triangles are similar. Explore More Use the diagram to complete each statement. 4 www.ck12.org Chapter 1. AA Similarity 1. 4SAM ∼ 4 SM ? 2. SA ? = ? = RI 3. SM = 4. T R = 5. 9? = 8? Answer questions 6-9 about trapezoid ABCD. 6. 7. 8. 9. 10. 11. Name two similar triangles. How do you know they are similar? Write a true proportion. Name two other triangles that might not be similar. If AB = 10, AE = 7, and DC = 22, find AC. Be careful! Writing How many angles need to be congruent to show that two triangles are similar? Why? Writing How do congruent triangles and similar triangles differ? How are they the same? Use the triangles below for questions 12-15. AB = 20, DE = 15, and BC = k. 5 www.ck12.org 12. 13. 14. 15. Are the two triangles similar? How do you know? Write an expression for FE in terms of k. If FE = 12, what is k? Fill in the blanks: If an acute angle of a _______ triangle is congruent to an acute angle in another ________triangle, then the two triangles are _______. Use the diagram below to answer questions 16-20. 16. Draw the three separate triangles in the diagram. 17. Explain why 4GDE ∼ 4DFE ∼ 4GFD. Complete the following proportionality statements. 18. 19. 20. GF DF GF GD GE DE = = = ? FE ? GE DE ? Answers for Explore More Problems To view the Explore More answers, open this PDF file and look for section 7.4. 6