4 5 Using Congruent Triangles Geometry Mrs Spitz

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4. 5 Using Congruent Triangles Geometry Mrs. Spitz Fall 2004

4. 5 Using Congruent Triangles Geometry Mrs. Spitz Fall 2004

Objectives: Use congruent triangles to plan and write proofs. Use congruent triangles to prove

Objectives: Use congruent triangles to plan and write proofs. Use congruent triangles to prove constructions are valid.

Assignment: Pgs. 232 -234 #1 -21 all

Assignment: Pgs. 232 -234 #1 -21 all

Planning a proof Knowing that all pairs of corresponding parts of congruent triangles are

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 ≅ RQS

Planning a proof For example, suppose you want to prove that PQS ≅ 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 corresponding parts of congruent triangles are congruent to conclude that PQS ≅ RQS.

Ex. 1: Planning & Writing a Proof Given: AB ║ CD, BC ║ DA

Ex. 1: Planning & Writing a Proof Given: AB ║ CD, BC ║ DA Prove: AB≅CD Plan for proof: Show that ∆ABD ≅ ∆CDB. Then use the fact that corresponding parts of congruent triangles are congruent.

Ex. 1: Planning & Writing a Proof Solution: First copy the diagram and mark

Ex. 1: Planning & Writing a Proof Solution: First copy the diagram and mark it with the 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 intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

Ex. 1: Paragraph Proof Because AD ║CD, it follows from the Alternate Interior Angles

Ex. 1: Paragraph Proof Because AD ║CD, it follows from the Alternate Interior Angles Theorem that ABD ≅ 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 ∆ABD ≅ ∆CDB. Finally because corresponding parts of congruent triangles are congruent, it follows that AB ≅ CD.

Ex. 2: Planning & Writing a Proof Given: A is the midpoint of MT,

Ex. 2: Planning & Writing a Proof Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Plan for proof: Prove that ∆MAS ≅ ∆TAR. Then use the fact that corresponding parts of congruent triangles are congruent to show that M ≅ T. Because these angles are formed by two segments intersected by a transversal, you can conclude that MS ║ TR.

Given: A is the midpoint of MT, A is the midpoint of SR. Prove:

Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Statements: Reasons: 1. 2. 3. 4. 5. 6. A is the midpoint of MT, A is the midpoint of SR. MA ≅ TA, SA ≅ RA MAS ≅ TAR ∆MAS ≅ ∆TAR M ≅ T MS ║ TR Given

Given: A is the midpoint of MT, A is the midpoint of SR. Prove:

Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Statements: Reasons: 1. Given 2. Definition of a midpoint 2. 3. 4. 5. 6. A is the midpoint of MT, A is the midpoint of SR. MA ≅ TA, SA ≅ RA MAS ≅ TAR ∆MAS ≅ ∆TAR M ≅ T MS ║ TR

Given: A is the midpoint of MT, A is the midpoint of SR. Prove:

Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Statements: Reasons: 1. Given 2. 3. Definition of a midpoint Vertical Angles Theorem 2. 3. 4. 5. 6. A is the midpoint of MT, A is the midpoint of SR. MA ≅ TA, SA ≅ RA MAS ≅ TAR ∆MAS ≅ ∆TAR M ≅ T MS ║ TR

Given: A is the midpoint of MT, A is the midpoint of SR. Prove:

Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Statements: Reasons: 1. Given 2. 3. 4. Definition of a midpoint Vertical Angles Theorem SAS Congruence Postulate 2. 3. 4. 5. 6. A is the midpoint of MT, A is the midpoint of SR. MA ≅ TA, SA ≅ RA MAS ≅ TAR ∆MAS ≅ ∆TAR M ≅ T MS ║ TR

Given: A is the midpoint of MT, A is the midpoint of SR. Prove:

Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Statements: Reasons: 1. Given 2. 3. 4. 5. Definition of a midpoint Vertical Angles Theorem SAS Congruence Postulate Corres. parts of ≅ ∆’s are ≅ 2. 3. 4. 5. 6. A is the midpoint of MT, A is the midpoint of SR. MA ≅ TA, SA ≅ RA MAS ≅ TAR ∆MAS ≅ ∆TAR M ≅ T MS ║ TR

Given: A is the midpoint of MT, A is the midpoint of SR. Prove:

Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Statements: Reasons: 1. Given 2. 3. 4. 5. 6. Definition of a midpoint Vertical Angles Theorem SAS Congruence Postulate Corres. parts of ≅ ∆’s are ≅ Alternate Interior Angles Converse. 2. 3. 4. 5. 6. A is the midpoint of MT, A is the midpoint of SR. MA ≅ TA, SA ≅ RA MAS ≅ TAR ∆MAS ≅ ∆TAR M ≅ T MS ║ TR

Ex. 3: Using more than one pair of triangles. Given: 1≅ 2, 3≅ 4.

Ex. 3: Using more than one pair of triangles. Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Plan for proof: The only information you have about ∆BCE and ∆DCE is that 1≅ 2 and that CE ≅CE. Notice, however, that sides BC and DC are also sides of ∆ABC and ∆ADC. If you can prove that ∆ABC≅∆ADC, you can use the fact that corresponding parts of congruent triangles are congruent to get a third piece of information about ∆BCE and ∆DCE. 2 1 3 4

Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Statements: 1. 2. 3. 4. 5. 6.

Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Statements: 1. 2. 3. 4. 5. 6. 1≅ 2, 3≅ 4 AC ≅ AC ∆ABC ≅ ∆ADC BC ≅ DC CE ≅ CE ∆BCE≅∆DCE 2 1 4 3 Reasons: 1. Given

Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Statements: 1. 2. 1≅ 2, 3≅ 4

Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Statements: 1. 2. 1≅ 2, 3≅ 4 AC ≅ AC 3. 4. 5. 6. ∆ABC ≅ ∆ADC BC ≅ DC CE ≅ CE ∆BCE≅∆DCE 2 1 4 3 Reasons: 1. 2. Given Reflexive property of Congruence

Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Statements: 2 1 4 3 Reasons: 1.

Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Statements: 2 1 4 3 Reasons: 1. 2. 1≅ 2, 3≅ 4 AC ≅ AC 1. 2. 3. ∆ABC ≅ ∆ADC 3. 4. 5. 6. BC ≅ DC CE ≅ CE ∆BCE≅∆DCE Given Reflexive property of Congruence ASA Congruence Postulate

Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Statements: 2 1 Reasons: 1. 2. 1≅

Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Statements: 2 1 Reasons: 1. 2. 1≅ 2, 3≅ 4 AC ≅ AC 1. 2. 3. ∆ABC ≅ ∆ADC 3. 4. 5. 6. BC ≅ DC CE ≅ CE ∆BCE≅∆DCE 4 3 4. Given Reflexive property of Congruence ASA Congruence Postulate Corres. parts of ≅ ∆’s are ≅

Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Statements: 2 1 Reasons: 1. 2. 1≅

Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Statements: 2 1 Reasons: 1. 2. 1≅ 2, 3≅ 4 AC ≅ AC 1. 2. 3. ∆ABC ≅ ∆ADC 3. 4. 5. 6. BC ≅ DC CE ≅ CE ∆BCE≅∆DCE 4 3 4. 5. Given Reflexive property of Congruence ASA Congruence Postulate Corres. parts of ≅ ∆’s are ≅ Reflexive Property of Congruence

Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Statements: 2 1 Reasons: 1. 2. 1≅

Given: 1≅ 2, 3≅ 4. Prove ∆BCE≅∆DCE Statements: 2 1 Reasons: 1. 2. 1≅ 2, 3≅ 4 AC ≅ AC 1. 2. 3. ∆ABC ≅ ∆ADC 3. 4. 5. 6. BC ≅ DC CE ≅ CE ∆BCE≅∆DCE 4 3 4. 5. 6. Given Reflexive property of Congruence ASA Congruence Postulate Corres. parts of ≅ ∆’s are ≅ Reflexive Property of Congruence SAS Congruence Postulate

Ex. 4: Proving constructions are valid In Lesson 3. 5 – you learned to

Ex. 4: 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. Then use the fact that corresponding

Plan for proof Show that ∆CAB ≅ ∆FDE. Then use the fact that corresponding parts of congruent triangles are congruent to conclude that CAB ≅ FDE. By construction, you can assume the following statements: – AB ≅ DE Same compass setting is used – AC ≅ DF Same compass setting is used – BC ≅ EF Same compass setting is used

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅ FDE 2

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅ FDE 2 1 4 3 Statements: Reasons: 1. 2. 3. 4. 5. 1. AB ≅ DE AC ≅ DF BC ≅ EF ∆CAB ≅ ∆FDE CAB ≅ FDE Given

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅ FDE 2

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅ FDE 2 1 4 3 Statements: Reasons: 1. 2. 3. 4. 5. 1. 2. AB ≅ DE AC ≅ DF BC ≅ EF ∆CAB ≅ ∆FDE CAB ≅ FDE Given

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅ FDE 2

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅ FDE 2 1 4 3 Statements: Reasons: 1. 2. 3. 4. 5. 1. 2. 3. AB ≅ DE AC ≅ DF BC ≅ EF ∆CAB ≅ ∆FDE CAB ≅ FDE Given

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅ FDE 2

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅ FDE 2 1 4 3 Statements: Reasons: 1. 2. 3. 4. 5. 1. 2. 3. 4. AB ≅ DE AC ≅ DF BC ≅ EF ∆CAB ≅ ∆FDE CAB ≅ FDE Given SSS Congruence Post

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅ FDE 2

Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB≅ FDE 2 1 4 3 Statements: Reasons: 1. 2. 3. 4. 5. AB ≅ DE AC ≅ DF BC ≅ EF ∆CAB ≅ ∆FDE CAB ≅ FDE Given SSS Congruence Post Corres. parts of ≅ ∆’s are ≅.

2 1 Given: QS RP, PT≅RT Prove PS≅ RS Statements: 1. 2. QS RP

2 1 Given: QS RP, PT≅RT Prove PS≅ RS Statements: 1. 2. QS RP PT ≅ RT 4 3 Reasons: 1. 2. Given