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ASA and AAS Triangle Congruence Dan Greenberg Lori Jordan Andrew Gloag Victor Cifarelli Jim Sconyers Bill Zahner 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: August 23, 2016 AUTHORS Dan Greenberg Lori Jordan Andrew Gloag Victor Cifarelli Jim Sconyers Bill Zahner www.ck12.org C HAPTER Chapter 1. ASA and AAS Triangle Congruence 1 ASA and AAS Triangle Congruence Here you’ll learn how to prove that triangles are congruent given information only about two of their angles and one of their sides. Angle-Side-Angle Postulate and Angle-Angle-Side Theorem If two angles and one side in one triangle are congruent to the corresponding two angles and one side in another triangle, then the two triangles are congruent. This idea encompasses two triangle congruence shortcuts: AngleSide-Angle and Angle-Angle-Side. Angle-Side-Angle (ASA) Congruence Postulate: If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the two triangles are congruent. Angle-Angle-Side (AAS) Congruence Theorem: If two angles and a non-included side in one triangle are congruent to two angles and the corresponding non-included side in another triangle, then the triangles are congruent. The placement of the word Side is important because it indicates where the side that you are given is in relation to the angles. The pictures below help to show the difference between the two shortcuts. ASA AAS What if you were given two triangles and provided with only the measure of two of their angles and one of their side lengths? How could you determine if the two triangles were congruent? MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/136751 1 www.ck12.org Examples Example 1 Can you prove that the following triangles are congruent? Why or why not? We cannot show the triangles are congruent because KLand ST are not corresponding, even though they are congruent. To determine if KLand ST are corresponding, look at the angles around them, 6 Kand 6 Land 6 Sand 6 T . 6 Khas one arc and 6 Lis unmarked. 6 Shas two arcs and 6 T is unmarked. In order to use AAS, 6 Sneeds to be congruent to 6 K. Example 2 Write a 2-column proof. Given: AB||ED, 6 C ∼ = 6 F, AB ∼ = ED Prove: AF ∼ = CD TABLE 1.1: Statement 1. AB||ED, 6 C ∼ = 6 F, AB ∼ = ED 6 2. 6 ABE ∼ DEB = 3. 4ABF ∼ = 4DEC 4. AF ∼ = CD Reason 1. Given 2. Alternate Interior Angles Theorem 3. ASA 4. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) Example 3 What information do you need to prove that these two triangles are congruent using the ASA Postulate, AB ∼ = UT , AC ∼ = UV , BC ∼ = TV , or 6 B ∼ = 6 T? 2 www.ck12.org Chapter 1. ASA and AAS Triangle Congruence For ASA, we need the side between the two given angles, which is AC and UV . The answer is AC ∼ = UV . Example 4 Write a 2-column proof. ∼ 6 E, AC = ∼ AE Given: 6 C = Prove: 4ACF ∼ = 4AEB TABLE 1.2: Statement 1. 6 C ∼ = 6 E, AC ∼ = AE ∼ 6 6 2. A = A 3. 4ACF ∼ = 4AEB Reason 1. Given 2. Reflexive PoC 3. ASA Example 5 What information do you need to prove that these two triangles are congruent using ASA? AAS? For ASA, we need the angles on the other side of EF and QR. 6 F ∼ =6 Q For AAS, we would need the other angle. 6 G ∼ =6 P 3 www.ck12.org Review For questions 1-3, determine if the triangles are congruent. If they are, write the congruence statement and which congruence postulate or theorem you used. 1. 2. 3. For questions 4-8, use the picture and the given information below. Given: DB ⊥ AC, DB is the angle bisector of 6 CDA 4. From DB ⊥ AC, which angles are congruent and why? 5. Because DB is the angle bisector of 6 CDA, what two angles are congruent? 6. From looking at the picture, what additional piece of information are you given? Is this enough to prove the two triangles are congruent? 7. Write a 2-column proof to prove 4CDB ∼ = 4ADB, using #4-6. ∼ 6 6 8. What would be your reason for C = A? For questions 9-13, use the picture and the given information. Given: LP||NO, LP ∼ = NO 4 www.ck12.org 9. 10. 11. 12. 13. Chapter 1. ASA and AAS Triangle Congruence From LP||NO, which angles are congruent and why? From looking at the picture, what additional piece of information can you conclude? Write a 2-column proof to prove 4LMP ∼ = 4OMN. What would be your reason for LM ∼ = MO? Fill in the blanks for the proof below. Use the given from above. Prove: M is the midpoint of PN. TABLE 1.3: Statement 1. LP||NO, LP ∼ = NO 2. 3. 4. LM ∼ = MO 5. M is the midpoint of PN. Reason 1. Given 2. Alternate Interior Angles 3. ASA 4. 5. Determine the additional piece of information needed to show the two triangles are congruent by the given postulate. 14. AAS 15. ASA 16. ASA 5 www.ck12.org 17. AAS Review (Answers) To see the Review answers, open this PDF file and look for section 4.8. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/1303 MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/1304 MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/1317 6