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ASA and AAS Triangle Congruence Andrew Gloag Bill Zahner Dan Greenberg Jim Sconyers Lori Jordan Victor Cifarelli 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®. 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Printed: November 21, 2012 AUTHORS Andrew Gloag Bill Zahner Dan Greenberg Jim Sconyers Lori Jordan Victor Cifarelli EDITOR Annamaria Farbizio www.ck12.org C ONCEPT Concept 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. 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? After completing this Concept, you’ll be able to use the Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) shortcuts to prove triangle congruency. Watch This MEDIA Click image to the left for more content. CK-12 ASA and AAS Triangle Congruence Watch the portions of the following two videos that deal with ASA and AAS triangle congruence. MEDIA Click image to the left for more content. James Sousa:Introduction toCongruent Triangles MEDIA Click image to the left for more content. James Sousa:DeterminingIf Two Triangles are Congruent Finally, watch this video. MEDIA Click image to the left for more content. James Sousa:Example2: Prove Two Triangles are Congruent 1 www.ck12.org Guidance 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 Example A What information do you need to prove that these two triangles are congruent using the ASA Postulate? a) AB ∼ = UT b) AC ∼ = UV c) BC ∼ = TV d) 6 B ∼ =6 T For ASA, we need the side between the two given angles, which is AC and UV . The answer is b. Example B Write a 2-column proof. 2 www.ck12.org Concept 1. ASA and AAS Triangle Congruence Given: 6 C ∼ = 6 E, AC ∼ = AE Prove: 4ACF ∼ = 4AEB TABLE 1.1: 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 C What information do you need to prove that these two triangles are congruent using: a) ASA? b) AAS? Solution: a) For ASA, we need the angles on the other side of EF and QR. 6 F ∼ =6 Q b) For AAS, we would need the other angle. 6 G ∼ =6 P MEDIA Click image to the left for more content. CK-12 ASA and AAS Triangle Congruence Guided Practice 1. Can you prove that the following triangles are congruent? Why or why not? 3 www.ck12.org 2. Write a 2-column proof. Given: BD is an angle bisector of 6 CDA, 6 C ∼ =6 A Prove: 4CBD ∼ = 6 ABD 3. Write a 2-column proof. Given: AB||ED, 6 C ∼ = 6 F, AB ∼ = ED Prove: AF ∼ = CD Answers: 1. We cannot show the triangles are congruent because KL and ST are not corresponding, even though they are congruent. To determine if KL and ST are corresponding, look at the angles around them, 6 K and 6 L and 6 S and 6 T . 6 K has one arc and 6 L is unmarked. 6 S has two arcs and 6 T is unmarked. In order to use AAS, 6 S needs to be congruent to 6 K. 2. TABLE 1.2: Statement 1. BD is an angle bisector of 6 CDA, 6 C ∼ =6 A ∼ 6 6 2. CDB = ADB 3. DB ∼ = DB 4. 4CBD ∼ = 4ABD 3. 4 Reason 1. Given 2. Definition of an Angle Bisector 3. Reflexive PoC 4. AAS www.ck12.org Concept 1. ASA and AAS Triangle Congruence TABLE 1.3: Statement 1. AB||ED, 6 C ∼ = 6 F, AB ∼ = ED ∼ 6 6 2. 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) Practice 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? 5 www.ck12.org 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 8. What would be your reason for 6 C ∼ A? = For questions 9-13, use the picture and the given information. Given: LP||NO, LP ∼ = NO 9. 10. 11. 12. 13. 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 1.4: 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 6 www.ck12.org Concept 1. ASA and AAS Triangle Congruence 16. ASA 17. AAS 7