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4.5 Using Congruent Triangles Objectives/Assignment Use congruent triangles to plan and write proofs Use congruent triangles to prove constructions are valid Assignment: 5-19 odd, 20, 21 Goal 1: Planning a Proof Knowing that all pairs of corresponding parts of congruent triangles are congruent can help you reach conclusions about congruent figures. Planning a Proof For example, suppose you want to prove that PQS ≅ Q RQS in the diagram shown at the right. One way to do this is to show that ∆PQS ≅ ∆RQS by the SSS Congruence Postulate. Then you can use the fact that R corresponding parts of P congruent triangles are congruent to conclude that S PQS ≅ RQS. Example 1: Planning & Writing a Proof Given: AB ║ CD, BC ║ AD B C Prove: AB≅CD Plan for proof: Show that ∆ABD ≅ ∆CDB. Then use the fact that corresponding parts of A D congruent triangles are congruent. Example 1: Planning & Writing a Proof Solution: First copy the diagram and mark it with the B C given information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments A D intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent. Example 1: Paragraph Proof Because AD ║CD, it follows from the Alternate Interior Angles Theorem that ABD B C ≅CDB. For the same reason, ADB ≅CBD because BC║DA. By the Reflexive property of Congruence, BD ≅ BD. You can use the ASA Congruence Postulate to conclude that A ∆ABD ≅ ∆CDB. Finally D because corresponding parts of congruent triangles are congruent, it follows that AB ≅ CD. Example 2: Planning & Writing a Proof Given: A is the midpoint of MT, A is the midpoint of SR. M R Prove: MS ║TR. Plan for proof: Prove that ∆MAS ≅ ∆TAR. Then use the fact that corresponding A parts of congruent triangles are congruent to show that M ≅ T. Because these T angles are formed by two S segments intersected by a transversal, you can conclude that MS ║ TR. M R Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. A S T Statements: Reasons: 1. A is the midpoint of MT, A 1. Given is the midpoint of SR. 2. MA ≅ TA, SA ≅ RA 2. Definition of a midpoint 3. MAS ≅ TAR 3. Vertical Angles Theorem 4. ∆MAS ≅ ∆TAR 4. SAS Congruence Postulate 5. M ≅ T 5. Corres. parts of ≅ ∆’s are ≅ 6. MS ║ TR 6. Alternate Interior Angles Converse Example 3: Using more than one pair of triangles Given: 1≅2, 3≅4 Prove ∆BCE≅∆DCE D Plan for proof: The only information you have about ∆BCE and ∆DCE is that 1≅2 2 4 and that CE ≅CE. Notice, C 3 A however, that sides BC and DC 1 E are also sides of ∆ABC and ∆ADC. If you can prove that ∆ABC≅∆ADC, you can use the B fact that corresponding parts of congruent triangles are congruent to get a third piece of information about ∆BCE and ∆DCE. Given: 1≅2, 3≅4. 2 4 Prove ∆BCE≅∆DCE C 1 E 3 A B Statements: Reasons: 1. 1≅2, 3≅4 1. Given 2. AC ≅ AC 2. Reflexive property of Congruence 3. ∆ABC ≅ ∆ADC 3. ASA Congruence Postulate 4. Corres. parts of ≅ ∆’s are ≅ 4. BC ≅ DC 5. Reflexive Property of 5. CE ≅ CE Congruence 6. ∆BCE≅∆DCE 6. SAS Congruence Postulate Goal 2: Proving Constructions are Valid In Lesson 3.5 you learned to copy an angle using a compass and a straight edge. The construction is summarized on pg. 159 and on pg. 231. Using the construction summarized above, you can copy CAB to form FDE. Write a proof to verify the construction is valid. Plan for Proof Show that ∆CAB ≅ ∆FDE. C Then use the fact that corresponding parts of congruent triangles are congruent to conclude that A CAB ≅ FDE. By construction, you can B assume the following F statements: – AB ≅ DE Same compass setting is used – AC ≅ DF Same compass D setting is used – BC ≅ EF Same compass E setting is used C Given: AB ≅ DE, AC ≅ DF, BC ≅ EF A Prove CAB≅FDE B F D E Statements: Reasons: 1. AB ≅ DE 1. Given 2. AC ≅ DF 2. Given 3. BC ≅ EF 3. Given 4. ∆CAB ≅ ∆FDE 4. SSS Congruence Post 5. CAB ≅ FDE 5. Corres. parts of ≅ ∆’s are ≅.

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posted: | 9/20/2011 |

language: | English |

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