Download Using Congruent Triangles

Triangle
Congruence: CPCTC
Geometry (Holt 4-7)
K. Santos
A
𝐶𝐴 bisects < DAB
D
B
C
How do you know the triangles are congruent?
SAS
Is < D ≅ < B?
yes
Why or why not?
They are corresponding parts of congruent triangles so they are
congruent
What other parts are congruent?
𝐷𝐶 ≅ 𝐵𝐶, < DCA ≅ < BCA
“Corresponding Parts of Congruent Triangles are
Congruent”
Comes from the definition of congruent triangles (or any figures)
The parts can be sides or angles.
CPCTC is an abbreviation for the phrase “corresponding parts of congruent
triangles are congruent”
Will use any time you know two triangles are congruent and then you
want to talk about some other corresponding parts (angles or sides) that
are congruent
It can be used as a justification (reason) in a proof after you have proven
two triangles are congruent.
Proof
Given: < Q ≅ < R
< QPS ≅ < RSP
Prove: 𝑆𝑄 ≅ 𝑃𝑅
Statements
1. < Q ≅ < R
2. < QPS ≅ < RSP
3. 𝑃𝑆 ≅ 𝑆𝑃
4. ∆ 𝑃𝑅𝑆 ≅ ∆𝑆𝑄𝑃
5. 𝑆𝑄 ≅ 𝑃𝑅
P
R
1.
2.
3.
4.
5.
Q
S
Reasons
Given
Given
Reflexive Property of
congruence
AAS Theorem (1, 2, 3)
Corresponding parts of
congruent triangles are
congruent
Proof
Given: 𝐾𝑀 bisects 𝐻𝐽
𝐾𝐻 ≅ 𝐾𝐽
Prove: < H ≅ < J
K
H
Statements
1. 𝐾𝑀 bisects 𝐻𝐽
2. M is a midpoint of 𝐻𝐽
3. 𝐻𝑀 ≅ 𝐽𝑀
4. 𝐾𝐻 ≅ 𝐾𝐽
5. 𝐾𝑀 ≅ 𝐾𝑀
6. ∆ 𝐾𝐻𝑀 ≅ ∆𝐾𝐽𝑀
7. < H ≅ < J
1.
2.
3.
4.
5.
6.
7.
M
J
Reasons
Given
Definition of a segment bisector
Definition of a midpoint
Given
Reflexive Property of congruence
SSS Postulate (3, 4, 5)
Corresponding parts of congruent
triangles are congruent