(4.4) CPCTC
Corresponding Parts of Congruent Triangles Congruent
Proving Triangles Congruent
LT: To use triangle congruence
and CPCTC to prove that parts
of two triangles are congruent.
Relevancy: This concept of CPCTC can be used measure
distance indirectly.
Proving Triangles Congruent
Can we use (SSS, SAS, AAS, or ASA) the given information to
prove triangles congruent?
ASA
SSS
AAS
Seriously?

Not
Possible
SAS
Proving Triangles Congruent
B
C
E
A
D
Given
Look For…
Reason
Midpoint
Two segments
are congruent
Defn. of midpoint
Two segments
are congruent
Defn. of bisector
Segment bisects
a segment
Example
If E is the midpoint of AC
then AE  CE
If AC bisects BD
then BE  DE

Two segments
parallel
Alt. int. angles
congruent
Alt. int. angles
theorem 
Vertical angles
Two congruent
angles
Vert. angles
theorem

If BA || CD
then A  C
and B  D
BEA  DEC
Proving Triangles Congruent
X
W
Given
Look For…
Segment bisects
an angle
Two angles
are congruent
Triangles that
share a side
A segment
congruent to itself
1 2
Z
Y
Reason
Example
Defn. of
angle bisector
Reflexive property

of congruence
If XZ bisects WXY
then WXZ  YXZ
XZ  XZ
If XZWY
Perpendicular
segments
Two angles
are congruent
1) Right angles by 
then 1, 2 are right 's
defn. of  lines
2) All right angles  then 1  2


Proving Triangles Congruent
With SSS, SAS, ASA, and AAS; you know how to
use three parts of triangles to show that the triangles
are congruent. Once you have triangles congruent,
you can make conclusions about their other parts
because by definition, Corresponding Parts of
Congruent Triangles are Congruent (CPCTC)
**You can use CPCTC in a proof.
Proving Triangles Congruent
X
Given : XZ bisects WXY
XZ  WY
Prove : WXZ  YXZ
W
Statement
Z
Reason
1. Given
2. XZ  WY
2. Given
3. Defn. of Angle Bisector
4. Right angles are
congruent because of defn.
of perpendicular lines
5. Reflexive Property of ≌
6. WXZ  YXZ
6. ASA
Y
Proving Triangles Congruent
B
C
Given : E is the midpoint of AC and BD
Prove : A  C
E
A
D
Statement
1. E is the midpoint of AC and BD
Reason
1. Given
2. Defn. of Midpoint
3.AEB
 CED
4. AED  CED
3. Vertical Angles
5. A  C
5. CPCTC
4. SAS
homework
1.
2.
3.
4.
5.
Sticker #3 Due: Wed. 1/29
Sticker #4 Due: Fri. 1/31
Practice WS (4.4) Due: Wed. 1/29
Study Island Challenge Due: Wed. 3/12
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