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. 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.)

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

Given HJ //LK, JK // HL Prove ∠ 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.

Given HJ //LK, JK // HL Prove ∠ 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.

Given HJ //LK, JK // HL Prove ∠ 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.

Given HJ //LK, JK // HL Prove ∠ 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.

Given HJ //LK, JK // HL Prove ∠ 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.

Given HJ //LK, JK // HL Prove ∠ 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.

Given HJ //LK, JK // HL Prove ∠ 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.

Given HJ //LK, JK // HL Prove ∠ 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.

Given HJ //LK, JK // HL Prove ∠ 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.

Given HJ //LK, JK // HL Prove ∠ 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.

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

Given: MS // TR, MS ≌ TR Prove: 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

Given: MS // TR, MS ≌ TR Prove: 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

Given: MS // TR, MS ≌ TR Prove: 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

Given: MS // TR, MS ≌ TR Prove: 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

Given: MS // TR, MS ≌ TR Prove: 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

Given: MS // TR, MS ≌ TR Prove: 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

Given: MS // TR, MS ≌ TR Prove: 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

Given: MS // TR, MS ≌ TR Prove: 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

Given: MS // TR, MS ≌ TR Prove: 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

Given: MS // TR, MS ≌ TR Prove: 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

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