Using Congruent Triangles Geometry Mrs. Spitz Fall 2004

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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