4-6
TriangleCongruence:
Congruence: CPCTC
CPCTC
4-6 Triangle
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
4-6 Triangle Congruence: CPCTC
Do Now
1. If ∆ABC  ∆DEF, then A 
2. If 1  2, why is a||b?
Holt Geometry
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and BC 
?
.
4-6 Triangle Congruence: CPCTC
Objective
Use CPCTC to prove parts of triangles
are congruent.
Holt Geometry
4-6 Triangle Congruence: CPCTC
Vocabulary
CPCTC
Holt Geometry
4-6 Triangle Congruence: CPCTC
CPCTC is an abbreviation for the phrase
“Corresponding Parts of Congruent
Triangles are Congruent.” It can be used
as a justification in a proof after you have
proven two triangles congruent.
Holt Geometry
4-6 Triangle Congruence: CPCTC
Remember!
SSS, SAS, ASA, AAS, and HL use
corresponding parts to prove triangles
congruent. CPCTC uses congruent
triangles to prove corresponding parts
congruent.
Holt Geometry
4-6 Triangle Congruence: CPCTC
Example 1: Engineering Application
A and B are on the edges
of a ravine. What is AB?
Holt Geometry
4-6 Triangle Congruence: CPCTC
Example 2
A landscape architect sets
up the triangles shown in
the figure to find the
distance JK across a pond.
What is JK?
Holt Geometry
4-6 Triangle Congruence: CPCTC
Example 3: Proving Corresponding Parts Congruent
Given: YW bisects XZ, XY  YZ.
Prove: XYW  ZYW
Statements
Reasons
1. 𝑌𝑊𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝑋𝑍
1. Given
2. 𝑋𝑌  𝑌𝑍
2. Given
3. 𝑋𝑊  𝑍𝑊
3. Def. segment bisector
4. 𝑌𝑊  𝑌𝑊
4. Reflexive POC
5. 𝑋𝑌𝑊 ZYW
5. SSS
6. 𝑋𝑌𝑊ZYW
6. CPCTC
Holt Geometry
Z
4-6 Triangle Congruence: CPCTC
Example 4
Given: PR bisects QPS and QRS.
Prove: PQ  PS
Statements
Reasons
1. 𝑃𝑅 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝑄𝑃𝑆 𝑎𝑛𝑑  QRS
1. Given
2. 𝑄𝑃𝑅SPR
2. Def.  bisector
3. 𝑄𝑅𝑃SRP
3. Def.  bisector
4. 𝑃𝑅  𝑃𝑅
4. Reflexive POC
5. 𝑄𝑃𝑅 SPR
5. ASA
6. 𝑃𝑄  𝑃𝑆
6. CPCTC
Holt Geometry
4-6 Triangle Congruence: CPCTC
Helpful Hint
Work backward when planning a proof. To
show that ED || GF, look for a pair of angles
that are congruent.
Then look for triangles that contain these
angles.
Holt Geometry
4-6 Triangle Congruence: CPCTC
Example 5: Using CPCTC in a Proof
Given: NO || MP, N  P
Prove: MN || OP
Statements
1
4
2
Reasons
1. 𝑁𝑂 𝑀𝑃
1. Given
2. 𝑁O
2. Given
3. 12
3. Alt. int.  th.
4. 𝑀𝑂  𝑀𝑂
4. Reflexive POC
5. 𝑁𝑂𝑀 PMO
5. AAS
6. 34
6. CPCTC
7. 𝑀𝑁 𝑂𝑃
7. Converse of alt. int.  th.
Holt Geometry
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4-6 Triangle Congruence: CPCTC
Example 6
Given: J is the midpoint of KM and NL.
Prove: KL || MN
3
1
Statements
Reasons
2
4
1. 𝐽 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐾𝑀 𝑎𝑛𝑑𝑁𝐿
1. Given
2. 𝐾𝐽  𝑀𝐽
2. Def. midpoint
3. 𝑁𝐽  𝐿𝐽
3. Def. midpoint
4. 12
4. Vertical  th.
5. 𝐾𝐽𝐿 MJN
5. SAS
6. 34
6. CPCTC
7. 𝐾𝐿 𝑀𝑁
7. Converse of alt. int.  th.
Holt Geometry
4-6 Triangle Congruence: CPCTC
You Try It!
Given: X is the midpoint of AC .
1  2
Prove: X is the midpoint of BD.
Statements
1. 𝑋 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐶
2. 12
Reasons
1. Given
2. Given
3. 𝐴𝑋  𝐶𝑋
3. Def. midpoint
4. 34
4. Vertical  th.
5. 𝐴𝑋𝐷 CXB
5. ASA
6. 𝐷𝑋  𝐵𝑋
6. CPCTC
7. 𝑋 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐵𝐷
7. Def. midpoint
Holt Geometry
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3