1 / 30

4.5 Using Congruent Triangles

4.5 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. Assignment:. Pgs. 232-234 #1-21 all. Planning a proof.

rana-langley
Download Presentation

4.5 Using Congruent Triangles

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 4.5 Using Congruent Triangles Geometry Mrs. Spitz Fall 2004

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

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

  4. Planning a proof • Knowing that all pairs of corresponding parts of congruent triangles are congruent can help you reach conclusions about congruent figures.

  5. 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. Planning a proof

  6. 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

  7. 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: Planning & Writing a Proof

  8. 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. 1: Paragraph Proof

  9. 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. Ex. 2: Planning & Writing a Proof

  10. Statements: A is the midpoint of MT, A is the midpoint of SR. MA ≅ TA, SA ≅ RA MAS ≅ TAR ∆MAS ≅ ∆TAR M ≅ T MS ║ TR Reasons: Given Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.

  11. Statements: A is the midpoint of MT, A is the midpoint of SR. MA ≅ TA, SA ≅ RA MAS ≅ TAR ∆MAS ≅ ∆TAR M ≅ T MS ║ TR Reasons: Given Definition of a midpoint Given: A is the midpoint of MT, A is themidpoint of SR.Prove: MS ║TR.

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

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

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

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

  16. 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. Ex. 3: Using more than one pair of triangles. 2 4 3 1

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

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

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

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

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

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

  23. 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.

  24. 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 Plan for proof

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

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

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

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

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

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

More Related