4.5 Using Congruent TrianglesC.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 // 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 ≌ 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

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

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

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

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

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

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

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

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

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

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