4.6 CPCTC Completed Notes

4.6
Use Congruent Triangles
CPCTC
Corresponding Parts of Congruent Triangles are Congruent
By definition, congruent triangles have congruent corresponding parts. So, if you can prove that two triangles are congruent, you know that their corresponding parts must be congruent as well.
When proving parts (angles and sides) of triangles congruent, you must first prove the triangles congruent by SSS, SAS, ASA, AAS or HL. Then ALL the parts will be congruent because:
Corresponding Parts of Congruent Triangles are Congruent.
This can be abbreviated CPCTC .
Ex 1: Tell which triangles you can show are congruent in order to prove the statement. What postulate or theorem would you use?
a.
b.
ΔABC ≅ ΔDBC by SSS; therefore, A ≅ D by CPCTC ΔQPR ≅ ΔTPS by SAS;
therefore, Q ≅ T by CPCTC Ex 1: Tell which triangles you can show are congruent in order to prove the statement. What postulate or theorem would you use?
c.
d.
ΔJKM ≅ ΔLKM by HL; ΔACD ≅ ΔDBA by AAS;
therefore, JM ≅ LM by CPCTC
therefore, AC ≅ DB by CPCTC
Ex 2: Use the diagram to write a plan for proof.
a.
1. SV ≅ UT 1. Given
2. ST ≅ UV
2. Given
3. VT ≅ TV
3. Reflexive POC
4. ΔSVT ≅ ΔUTV
4. SSS
5. S ≅ U
5. CPCTC
Ex 3: Use the diagram to write a plan for proof.
b.
1. NLM and PLQ
are vertical 's 1. Definition of vertical
angles
2. NLM ≅ PLQ 2. Vertical Angle Congruence Theorem
3. N ≅ P 3. Given
4. NM ≅ PQ
4. Given
5. ΔNLM ≅ ΔPLQ 5. AAS
6. LM ≅ LQ
6. CPCTC
Ex 4: Given: Prove: 1. YX ≅ WX
1. Given
2. ZX bisects YXW
2. Given
3. YXO ≅ WXO
3. Definition of angle bisector
4. XO ≅ XO
4. Reflexive POC
5. ΔYXO ≅ ΔWXO
5. SAS
6. YO ≅ WO
6. CPCTC
Ex 5: Given:
Prove: D is the midpoint of 1. AD BC
1. Given
2. BDA & CDA are right 's
2. Definition of perpendicular
3. ΔBDA & ΔCDA are right
3. Definition of right triangles
triangles 4. AB ≅ AC
4. Given
5. AD ≅ AD
5. Reflexive POC
6. ΔBDA ≅ ΔCDA
6. HL
7. BD ≅ CD
7. CPCTC
8. D is the midpoint of BC
8. Definition of midpoint

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