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4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Do Now Lesson Presentation Exit Ticket 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Warm Up #2 1. What are sides AC and BC called? Side AB? legs; hypotenuse 2. Which side is in between A and C? AC 3. Given DEF and GHI, if D G and E H, why is F I? Third s Thm. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Know: Solve It ! Given: 𝐴𝐵 ∥ 𝐶𝐷 and 𝐴𝐷 ∥ 𝐶𝐵 Prove: ∆ABC ∆CDA 1. What do you need to find to solve the problem ? at least three corresponding pairs of sides or angles that we can prove to be congruent y◦ 2. What are the corresponding parts of the two triangles ? D and B; DAC and BCA; CAB and ACD; 𝐴𝐶 and 𝐶𝐴; 𝐴𝐷 and 𝐶𝐵; 𝐴𝐵 and 𝐶𝐷 3. What word would you use to describe 𝑨𝑪 ? transversal 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Communicate: Connect Mathematical Ideas (1)(F) Given: 𝐴𝐵 ∥ 𝐶𝐷 and 𝐴𝐷 ∥ 𝐶𝐵 Prove: ∆ABC ∆CDA 4. What can you show about angles in the triangles that can indicate congruency? We can find congruent angles using alternate interior angles of y◦ the transversal. 5. What do you know about a side or sides of the triangles that can be used to show congruency? The transversal is part of both triangles, so it is congruent to itself by the Reflexive Property of Congruence. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Connect to Math By the end of today’s lesson, SWBAT 1. Apply ASA, AAS to construct triangles and to solve problems. 2. Prove triangles congruent by using ASA, AAS, and Applying CPCTC. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Vocabulary included side CPCTC 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Essential Understanding You can prove that two triangles are congruent without having to show that all corresponding parts are congruent. In this lesson, you will prove triangles congruent by using one pair of corresponding sides and two pairs of corresponding angles. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Example 1: Applying ASA Congruence Which two triangles are congruent by ASA ? Explain. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Example 1: Applying ASA Congruence Which two triangles are congruent by ASA ? Explain. In ∆SUV, 𝑈𝑉 included between U and V and has a congruence marking. In ∆NEO, 𝐸𝑂 included between E and O and has a congruence marking. In ∆ATW, 𝑇𝑊 included between T and W but does not have congruence marking. Since U ≌ E , 𝑈𝑉 ≅ 𝐸𝑂, and V ≌ O ; therefore, ∆SUV ≌ ∆NEO. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Example 2: Applying ASA Congruence Determine if you can use ASA to prove the triangles congruent. Explain. Two congruent angle pairs are give, but the included sides are not given as congruent. Therefore ASA cannot be used to prove the triangles congruent. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Example 3 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. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Example 4: Angle-Side-Angle Congruence Statements Reasons 1. G K; J M 1. Given 2. H L 2. Third s Thrm. 3. 𝐻𝐽 𝐿𝑀 3. Given 4. ∆GHJ ∆KLM 4. ASA Steps 1, 3, 2 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Example 5: Writing a Proof Recreation Members of a teen organization are building a miniature golf course at your town’s youth center. The design plan calls for the first hole to have two congruent triangular bumpers. Prove that the bumpers on the first hole, shown at the right, meet the conditions of the plan. Given: 𝐴𝐵 ≅ 𝐷𝐸, A ≌ D, B and E are right angles Prove: ∆ABC ≅ ∆DEF Proof: It is given that A ≌ D and 𝐴𝐵 ≅ 𝐷𝐸. B ≌ E because all rights angles are congruent. Hence, ∆ABC ≅ ∆DEF by ASA. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side (AAS). 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Example 6: Writing a Proof Using AAS Given: M ≌ K , 𝑊𝑀 ∥ 𝑅𝐾, Prove: ∆WMR ≅ ∆RKW Statements Reasons 1. M K 2. 𝑊𝑀 || 𝑅𝐾 1. Given 2. Given 3. MWR KRW 3. Alt. Int. s Thm. 4. 𝑊𝑅 𝑊𝑅 5. ∆WMR ∆RKW 4. Reflex. Prop. of 5. AAS Steps 1, 3, 4 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Example 7: Writing a Proof Using AAS Given: 𝐽𝐿 bisects KLM, K M Prove: JKL JML 𝐽𝐿 bisects KLM Given KLJ ≌ MLJ Def. of bisector K ≌ M 𝐽𝐿 ≌ 𝐽𝐿 Given Reflexive Prop. of ≌ JKL JML AAS 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles congruent. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Remember ! SSS, SAS, ASA, and AAS use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Example 8: Writing a Proof Using CPCTC Given: M ≌ K , 𝑊𝑀 ∥ 𝑅𝐾, Prove: 𝑀𝑅 ≅ 𝐾𝑊 Statements Reasons 1. M K 1. Given 2. 𝑊𝑀 || 𝑅𝐾 3. MWR KRW 4. 𝑊𝑅 𝑊𝑅 5. ∆WMR ∆RKW 2. Given 3. Alt. Int. s Thm. 4. Reflex. Prop. of 6. 𝑀𝑅 ≅ 𝐾𝑊 6. CPCTC 5. AAS Steps 1, 3, 4 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Example 9: Landscaping Application 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. 4-3 Triangle Congruence: ASA, AAS, and Applying CPCTC Exit Ticket Identify the postulate or theorem that proves the triangles congruent. 2. Given: FAB GED, ABC DCE, AC EC Prove: BC DC