Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
At the beginning of this chapter, we discussed that corresponding parts of congruent triangles are congruent.
In this lesson, instead of focusing on proving that two triangles are congruent, we will use that fact that they are congruent, to prove
that their pieces are congruent.
Let’s assume that PTR ≡ PSQ. By CPCTC, we know that:
PT ≡ PS
TR ≡ SQ
PR ≡ PQ
<S ≡ <T
<Q ≡ <R
M
Which two triangles would have to be congruent in order to show that:
1. SM ≡ MT?
3. SR ≡ TQ
2. <RSQ ≡ <QTR?
4. <MTQ ≡ <MSR
The next set of proofs you will learn are CPCTC proofs. In them, you will use the fact that triangles are congruent, in order to show
that their parts are congruent.
The “givens” in these proofs will provide enough information to show that a pair of triangles is congruent.
Once you have shown that two triangles are congruent, you can use that information to show that each of the segments in that triangle
are congruent; each of the angles in that triangle are congruent; and in some cases, that certain lines are parallel.
On the next page are a few examples:
Level 1
Given:
PS ≡ SR and PQ ≡ QR
Prove:
<PQS ≡ <RQS
Q
R
P
Statements
PQ ≡ QR
QS ≡ QS
PS ≡ SR
PQS ≡ RQS
<PQS ≡ <RQS
If we can show that PQS ≡
RQS, we can show that <PQS ≡
<RQS because they are
corresponding angles in the two
triangles
Reasons
Given
Reflexive Property
Given
SSS Congruence Postulate
CPCTC
S
Level 2
Given: A is the midpoint of MT and SR
Prove: MS || RT
M
R
A
S
T
Statements
A is the midpoint of MT
MA ≡ AT
<MAS ≡ <TAR
A is the midpoint of SR
SA ≡ AR
MAS ≡ TAR
<R ≡ <S
MS || RT
---
Reasons
Given
Definition of Midpoint
Vertical Angles
Given
Definition of Midpoint
SAS
CPCTC
Alternate Interior Angles Converse
Sometimes the plan for proof can be particularly tricky to see; it can helpful to think backwards.
Level 3
Given: PA ≡ AK ; AL ≡ AN
In order to show that AX ≡ AY,
we need to show that AXL ≡
AYN. The “givens” don’t give
us enough info to prove those two
triangles are congruent, so we
will have to prove that the larger
triangles are congruent first
Prove: AX ≡ AY
Try to complete this proof on your own:
Given: <1 ≡ <2 ; <5 ≡ <6
Prove: BCO ≡ DCO
Statements
<1 ≡ <2
AC ≡ AC
Reasons
Given
<5 ≡ <6
BC ≡ CD
Given
<5 ≡ <6
OC ≡ OC
Given
BCO ≡ DCO
Statements
PA ≡ AK
<PAL ≡ < NAK
AL ≡ AN
PAL ≡ NAK
<N ≡ <L
AL ≡ AN
<XAL ≡ <NAY
AXL ≡ AYN
AX ≡ AY
Reasons
Given
Vertical Angles
Given
SAS Congruence
CPCTC
Given
Vertical Angles
ASA Congruence
CPCTC