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5.3 Proving Triangles are Congruent – ASA & AAS Objectives: • Show triangles are congruent using ASA and AAS. Key Vocabulary Included Side Postulates 14 Angle–Side–Angle (ASA) Congruence Postulate Theorems 5.1 Angle-Angle-Side (AAS) Congruence Theorem Definition: Included Side An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side. Example: Included Sided C The side between 2 angles A B Y X INCLUDED SIDE Z Postulate 14 (ASA): Angle-SideAngle Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent. Angle-Side-Angle (ASA) Congruence Postulate Two angles and the INCLUDED side Example 1 Determine When To Use ASA Congruence Based on the diagram, can you use the ASA Congruence Postulate to show that the triangles are congruent? Explain your reasoning. a. b. SOLUTION a. You are given that C E, B F, and BC FE. You can use the ASA Congruence Postulate to show that ∆ABC ∆DFE. b. You are given that R Y and S X. You know that RT YZ, but these sides are not included between the congruent angles, so you cannot use the ASA Congruence Postulate. Example 2: Applying ASA Congruence Determine if you can use ASA to prove the triangles congruent. Explain. Two congruent angle pairs are given, but the included sides are not given as congruent. Therefore ASA cannot be used to prove the triangles congruent. Your Turn Determine if you can use ASA to prove NKL LMN. Explain. By the Alternate Interior Angles Theorem. KLN MNL. NL LN by the Reflexive Property. No other congruence relationships can be determined, so ASA cannot be applied. Theorem 5.1 (AAS): Angle-AngleSide Congruence Theorem If two angles and a non-included side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the triangles are congruent. AAS A B D C F E OR X Y H Z I J Angle-Angle-Side (AAS) Congruence Theorem Two Angles and One Side that is NOT included Example 3 Determine What Information is Missing What additional congruence is needed to show that ∆JKL ∆NML by the AAS Congruence Theorem? SOLUTION You are given KL ML. Because KLJ and MLN are vertical angles, KLJ MLN. The angles that make KL and ML the non-included sides are J and N, so you need to know that J N. Example 4 Decide Whether Triangles are Congruent Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. b. a. c. SOLUTION a. EF JH E J Given FGE HGJ Vertical Angles Theorem Given Use the AAS Congruence Theorem to conclude that ∆EFG ∆JHG. Example 4 Decide Whether Triangles are Congruent b. c. b. Based on the diagram, you know only that MP QN and NP NP. You cannot conclude that the triangles are congruent. c. UZW XWZ Alternate Interior Angles Theorem WZ WZ Reflexive Prop. of Congruence UWZ XZW Alternate Interior Angles Theorem Use the ASA Congruence Postulate to conclude that ∆WUZ ∆ZXW. Example 5: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. Example 6: In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. Thus, you can use the AAS Congruence Theorem to prove that ∆EFG ∆JHG. Example 7: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. Example 8: In addition to the congruent segments that are marked, NP NP. Two pairs of corresponding sides are congruent. This is not enough information (CBD) to prove the triangles are congruent. Example 9 Prove Triangles are Congruent A step in the Cat’s Cradle string game creates the triangles shown. Prove that ∆ABD ∆EBC. A SOLUTION C B BD BC, AD || EC D ∆ABD ∆EBC Statements Reasons 1. BD BC 1. Given 2. AD || EC 2. Given 3. D C 3. Alternate Interior Angles Theorem 4. ABD EBC 4. Vertical Angles Theorem 5. ∆ABD ∆EBC 5. ASA Congruence Postulate E Your Turn: 1. Complete the statement: You can use the ASA Congruence Postulate when the congruent sides are _____ ? between the corresponding congruent angles. ANSWER included Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. 3. 2. ANSWER no ANSWER 4. no ANSWER yes; AAS Congruence Theorem Congruence Shortcuts } Ways To Prove Triangles Are Congruent Congruence Shortcuts AAA and SSA??? Does AAA and SSA provide enough information to determine the exact shape and size of a triangle? AAA and SSA??? Does AAA and SSA provide enough information to determine the exact shape and size of a triangle? NO Not Congruence Shortcuts } NO BAD WORDS Do Not prove Triangle Congruence NO CAR INSURANCE Triangle Congruence Practice Your Turn Is it possible to prove the Δs are ? ( No, there is no AAA ! CBD Yes, ASA Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write cannot be determined (CBD). G K I H J ΔGIH ΔJIK by AAS In ΔDEF and ΔLMN , D N , DE NL and E L. Write a congruence statement. DEF NLM ASA by ____ What other pair of angles needs to be marked so that the two triangles are congruent by AAS? D L E N M F E N What other pair of angles needs to be marked so that the two triangles are congruent by ASA? D L D L M F E N Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write cannot be determined (CBD). E A C B D ΔACB ΔECD by SAS Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write cannot be determined (CBD). J M K L ΔJMK ΔLKM by SAS or ASA Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write cannot be determined (CBD). J T L K V U Cannot Be Determined (CBD) BC YZ or AC XZ B Y A X Cannot Be Determined (CBD) – SSA is not a valid Congruence Shortcut. Yes, ∆TNS ≅ ∆UHS by AAS Review Remember! SSS, SAS, ASA, and AAS use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. Example 10: Using CPCTC A and B are on the edges of a ravine. What is AB? One angle pair is congruent, because they are vertical angles. Two pairs of sides are congruent, because their lengths are equal. Therefore the two triangles are congruent by SAS. By CPCTC, the third side pair is congruent, so AB = 18 mi. Your Turn A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles. Two pairs of sides are congruent, because their lengths are equal. Therefore the two triangles are congruent by SAS. By CPCTC, the third side pair is congruent, so JK = 41 ft. Joke Time Which one came first the egg or the chicken? I don't care I just want my breakfast served. What do you call a handsome intelligent sensitive man? A rumor. What does a clock do when it's hungry? Goes back 4 secounds!!! Assignment Pg. 253 - 256 #1 – 21 odd, 25 – 45 odd