### 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.
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
∠ HJL ≌ ∠ KLJ
#3. //, Alt. Int. ⦞ ≌
#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
∠ HJL ≌ ∠ KLJ
#3. //, Alt. Int. ⦞ ≌
#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
#3.
∠ HLJ ≌ ∠ KJL
∠ HJL ≌ ∠ KLJ
#3. //, Alt. Int. ⦞ ≌
#4.
⍙LHJ ≌ ⍙ JKL
#4.
A.S.A.
#5.
∠ LHJ ≌ ∠ JKL
#5.
C.P.C.T.
#2. Reflexive
Given HJ //LK, JK // HL
Prove ∠ LHJ ≌ ∠ JKL
#1.
HJ // LK, JK // HL
#1.Given
#2.
LJ ≌ LJ
#2. Reflexive
#3.
∠ HLJ ≌ ∠ KJL
∠ HJL ≌ ∠ KLJ
#3. //, Alt. Int. ⦞ ≌
#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
∠ HJL ≌ ∠ KLJ
#3. //, Alt. Int. ⦞ ≌
#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
∠ HJL ≌ ∠ KLJ
#3. //, Alt. Int. ∠s ≌
#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
∠ HJL ≌ ∠ KLJ
#3. //, Alt. Int. ∠s ≌
#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
∠ HJL ≌ ∠ KLJ
# 3. //, Alt. Int. ∠s ≌
#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
∠ HJL ≌ ∠ KLJ
#3. //, Alt. Int. ∠s ≌
#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
∠ HJL ≌ ∠ KLJ
#3. //, Alt. Int. ∠s ≌
#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
∠S ≌ ∠ R
#2.
// , Alt. Int. ⦞ ≌
#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
∠S ≌ ∠ R
#2.
// , Alt. Int. ⦞ ≌
#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
∠S ≌ ∠ R
#2.
// , Alt. Int. ⦞ ≌
#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
∠S ≌ ∠ R
#2.
// , Alt. Int. ∠s ≌
#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
∠S ≌ ∠ R
#2.
// , Alt. Int. ∠s ≌
#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
∠S ≌ ∠ R
#2.
// , Alt. Int. ∠s ≌
#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
∠S ≌ ∠ R
#2.
// , Alt. Int. ∠s ≌
#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
∠S ≌ ∠ R
#2.
// , Alt. Int. ∠s ≌
#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
∠S ≌ ∠ R
#2.
// , Alt. Int. ∠s ≌
#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
∠S ≌ ∠ R
#2.
// , Alt. Int. ∠s ≌
#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
HOMEWORK
Page 232 – 235
# 4 – 10, 13, 14,
18, 22, 25 - 36
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