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4.5 Using Congruent Triangles

4.5 Using Congruent Triangles. C.P.C.T. . The definition of congruent . Same shape and same size. If we have a congruent triangles, then all the matching parts are congruent.

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4.5 Using Congruent Triangles

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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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