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Understanding Congruent Triangles: C.P.C.T. and Proofs

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This resource provides a concise overview of congruent triangles and the concept of Corresponding Parts of Congruent Triangles (C.P.C.T.). It explains that congruence implies equal shape and size, leading to matching angles and sides being congruent. The text illustrates the use of C.P.C.T. in geometric proofs, including examples of parallel lines and corresponding angles. Additionally, it covers the midpoint theorem and offers a structured proof to demonstrate how the properties of congruent triangles can be applied in real scenarios.

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Understanding Congruent Triangles: C.P.C.T. and Proofs

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  1. 4.5 Using Congruent Triangles C.P.C.T.

  2. The definition of congruent Same shape and same size. If we have a congruent triangles, then all the matching parts are congruent. Congruent Parts of Congruent Triangles are Congruent. (C.P.C.T.) if you want to add another C that would be fine. (C.P.C.T.C.)

  3. Given that ∠ ABC≌ ∠ XYZ Then ∠ ABC ≌ ∠ XYZ by C.P.C.T. We no longer need to say by the definition of Congruent triangles. We will use it in the next proof

  4. Given HJ //LK, JK // HLProve ∠ LHJ ≌∠ JKL #1. HJ // LK, JK // HL #1.Given #2. LJ ≌ LJ #2. Reflexive #3. ∠ HLJ ≌ ∠ KJL #3. //, Alt. Int. ⦞ ≌ ∠ HJL ≌ ∠ KLJ #4. ⍙LHJ ≌ ⍙ JKL #4. A.S.A. #5. ∠ LHJ ≌ ∠ JKL #5. C.P.C.T.

  5. Given HJ //LK, JK // HLProve ∠ LHJ ≌∠ JKL #1. HJ // LK, JK // HL #1.Given #2. LJ ≌ LJ #2. Reflexive #3. ∠ HLJ ≌ ∠ KJL #3. //, Alt. Int. ⦞ ≌ ∠ HJL ≌ ∠ KLJ #4. ⍙LHJ ≌ ⍙ JKL #4. A.S.A. #5. ∠ LHJ ≌ ∠ JKL #5. C.P.C.T.

  6. Given HJ //LK, JK // HLProve ∠ LHJ ≌∠ JKL #1. HJ // LK, JK // HL #1.Given #2. LJ ≌ LJ #2. Reflexive #3. ∠ HLJ ≌ ∠ KJL #3. //, Alt. Int. ⦞ ≌ ∠ HJL ≌ ∠ KLJ #4. ⍙LHJ ≌ ⍙ JKL #4. A.S.A. #5. ∠ LHJ ≌ ∠ JKL #5. C.P.C.T.

  7. Given HJ //LK, JK // HLProve ∠ LHJ ≌∠ JKL #1. HJ // LK, JK // HL #1.Given #2. LJ ≌ LJ #2. Reflexive #3. ∠ HLJ ≌ ∠ KJL #3. //, Alt. Int. ⦞ ≌ ∠ HJL ≌ ∠ KLJ #4. ⍙LHJ ≌ ⍙ JKL #4. A.S.A. #5. ∠ LHJ ≌ ∠ JKL #5. C.P.C.T.

  8. Given HJ //LK, JK // HLProve ∠ LHJ ≌∠ JKL #1. HJ // LK, JK // HL #1.Given #2. LJ ≌ LJ #2. Reflexive #3. ∠ HLJ ≌∠ KJL #3. //, Alt. Int. ⦞ ≌ ∠ HJL ≌∠ KLJ #4. ⍙LHJ ≌ ⍙ JKL #4. A.S.A. #5. ∠ LHJ ≌ ∠ JKL #5. C.P.C.T.

  9. Given HJ //LK, JK // HLProve ∠ LHJ ≌∠ JKL #1. HJ // LK, JK // HL #1.Given #2. LJ ≌ LJ #2. Reflexive #3. ∠ HLJ ≌∠ KJL #3. //, Alt. Int. ∠s≌ ∠ HJL ≌∠ KLJ #4. ⍙LHJ ≌ ⍙ JKL #4. A.S.A. #5. ∠ LHJ ≌ ∠ JKL #5. C.P.C.T.

  10. Given HJ //LK, JK // HLProve ∠ LHJ ≌ ∠ JKL #1. HJ // LK, JK // HL #1.Given #2. LJ ≌ LJ #2. Reflexive #3. ∠ HLJ ≌∠ KJL #3. //, Alt. Int. ∠s≌∠ HJL ≌∠ KLJ #4. ⍙LHJ ≌ ⍙ JKL#4. A.S.A. #5. ∠ LHJ ≌ ∠ JKL #5. C.P.C.T.

  11. Given HJ //LK, JK // HLProve ∠ LHJ ≌∠ JKL #1. HJ // LK, JK // HL #1.Given #2. LJ ≌ LJ #2. Reflexive #3. ∠ HLJ ≌∠ KJL # 3. //, Alt. Int. ∠s≌ ∠ HJL ≌∠ KLJ #4. ΛLHJ ≌Λ JKL #4. A.S.A. #5. ∠ LHJ ≌ ∠ JKL #5. C.P.C.T.

  12. Given HJ //LK, JK // HLProve ∠ LHJ ≌∠ JKL #1. HJ // LK, JK // HL #1.Given #2. LJ ≌ LJ #2. Reflexive #3. ∠ HLJ ≌∠ KJL #3. //, Alt. Int. ∠s≌∠ HJL ≌∠ KLJ #4. ΛLHJ ≌Λ JKL #4. A.S.A. #5. ∠ LHJ ≌∠ JKL #5. C.P.C.T.

  13. Given HJ //LK, JK // HLProve ∠ LHJ ≌∠ JKL #1. HJ // LK, JK // HL #1.Given #2. LJ ≌ LJ #2. Reflexive #3. ∠ HLJ ≌∠ KJL #3. //, Alt. Int. ∠s≌ ∠ HJL ≌∠ KLJ #4. ΛLHJ ≌ΛJKL #4. A.S.A. #5. ∠ LHJ ≌∠ JKL #5. C.P.C.T.

  14. We can parts of congruent triangles to prove other things in the triangle. In the next proof, we will look at this idea

  15. Given: MS // TR, MS ≌ TRProve: A is the midpoint of MT #1. MS // TR, MS ≌ TR #1. Given #2. ∠M ≌ ∠T #2. // , Alt. Int. ⦞ ≌ ∠S ≌ ∠ R #3. ⍙MAS ≌⍙TAR #3. A.S.A. #4. MA ≌ TA #4. C.P.C.T. #5. A is the midpoint of MT. #5. Def. of midpoint

  16. Given: MS // TR, MS ≌ TRProve: A is the midpoint of MT #1. MS // TR, MS ≌ TR #1. Given #2. ∠M ≌ ∠T #2. // , Alt. Int. ⦞ ≌ ∠S ≌ ∠ R #3. ⍙MAS ≌⍙TAR #3. A.S.A. #4. MA ≌ TA #4. C.P.C.T. #5. A is the midpoint of MT. #5. Def. of midpoint

  17. Given: MS // TR, MS ≌ TRProve: A is the midpoint of MT #1. MS // TR, MS ≌ TR #1. Given #2. ∠M ≌ ∠T #2. // , Alt. Int. ⦞ ≌ ∠S ≌ ∠ R #3. ⍙MAS ≌⍙TAR #3. A.S.A. #4. MA ≌ TA #4. C.P.C.T. #5. A is the midpoint of MT. #5. Def. of midpoint

  18. Given: MS // TR, MS ≌ TRProve: A is the midpoint of MT #1. MS // TR, MS ≌ TR #1. Given #2. ∠M ≌ ∠T #2. // , Alt. Int. ∠s ≌ ∠S ≌ ∠ R #3. ⍙MAS ≌⍙TAR #3. A.S.A. #4. MA ≌ TA #4. C.P.C.T. #5. A is the midpoint of MT. #5. Def. of midpoint

  19. Given: MS // TR, MS ≌ TRProve: A is the midpoint of MT #1. MS // TR, MS ≌ TR #1. Given #2. ∠M ≌ ∠T #2. // , Alt. Int. ∠s ≌ ∠S ≌ ∠ R #3. ∠ MAS ≌ ∠ TAR #3. A.S.A. #4. MA ≌ TA #4. C.P.C.T. #5. A is the midpoint of MT. #5. Def. of midpoint

  20. Given: MS // TR, MS ≌ TRProve: A is the midpoint of MT #1. MS // TR, MS ≌ TR #1. Given #2. ∠M ≌ ∠T #2. // , Alt. Int. ∠s ≌ ∠S ≌ ∠ R #3. ∠ MAS ≌ ∠ TAR #3. A.S.A. #4. MA ≌ TA #4. C.P.C.T. #5. A is the midpoint of MT. #5. Def. of midpoint

  21. Given: MS // TR, MS ≌ TRProve: A is the midpoint of MT #1. MS // TR, MS ≌ TR #1. Given #2. ∠M ≌ ∠T #2. // , Alt. Int. ∠s ≌ ∠S ≌ ∠ R #3. ∠ MAS ≌ ∠ TAR #3. A.S.A. #4. MA ≌ TA #4. C.P.C.T. #5. A is the midpoint of MT. #5. Def. of midpoint

  22. Given: MS // TR, MS ≌ TRProve: A is the midpoint of MT #1. MS // TR, MS ≌ TR #1. Given #2. ∠M ≌ ∠T #2. // , Alt. Int. ∠s ≌ ∠S ≌ ∠ R #3. ∠ MAS ≌ ∠ TAR #3. A.S.A. #4. MA ≌ TA #4. C.P.C.T. #5. A is the midpoint of MT. #5. Def. of midpoint

  23. Given: MS // TR, MS ≌ TRProve: A is the midpoint of MT #1. MS // TR, MS ≌ TR #1. Given #2. ∠M ≌ ∠T #2. // , Alt. Int. ∠s ≌ ∠S ≌ ∠ R #3. ∠ MAS ≌ ∠ TAR #3. A.S.A. #4. MA ≌ TA #4. C.P.C.T. #5. A is the midpoint of MT. #5. Def. of midpoint

  24. Given: MS // TR, MS ≌ TRProve: A is the midpoint of MT #1. MS // TR, MS ≌ TR #1. Given #2. ∠M ≌ ∠T #2. // , Alt. Int. ∠s ≌ ∠S ≌ ∠ R #3. ∠ MAS ≌ ∠ TAR #3. A.S.A. #4. MA ≌ TA #4. C.P.C.T. #5. A is the midpoint of MT. #5. Def. of midpoint

  25. Page 232 – 235 # 4 – 10, 13, 14, 18, 22, 25 - 36 Homework

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