CPCTC in Proofs

You should be familiar that if two triangles are congruent, then all their corresponding parts are congruent as well.

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​CPCTC: Corresponding Parts of Congruent Triangles are Congruent

Multiple Choice

What does CPCTC stand for?

• Congruent parts of congruent triangles are congruent

• Corresponding parts of congruent triangles are congruent

• Corresponding parts of corresponding triangles are corresponding

• Corresponding parts of congruent triangles are Canadian.

Multiple Choice

Remember order matters. Which of the following is true?

• ∆FGH ≅ ∆VHW

• ∆HFG ≅ ∆HWV

• ∆HGF ≅ ∆VHW

• ∆FGH ≅ ∆VWH

Multiple Choice

If ∆LMK ≅ ∆LMT, then
∠K ≅ ____

• T

• ∠T

• ∠M

• ∠L

Multiple Choice

∆ABC≅∆XYZ
which is true?

• AB≅XY

• AB≅YX

• AB≅XZ

• BC≅CB

Multiple Choice

What is the correct congruence statement?  ΔDEC ≅________

• ΔAEB

• ΔEAB

• ΔBEA

• Not Congruent

To prove CPCTC:

First, we need to prove that the two triangles are congruent with the help of any one of the triangle congruence criteria.

In the figure, determine how you could prove the triangles congruent.

SSS, SAS, AAS, ASA, HL

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Multiple Choice

What reason could prove the triangles congruent?

• SSS

• SAS

• AAS

• ASA

• HL

Hopefully you recognize

• BC≅CD

• Vertical angles are congruent

• AC≅EC

This is enough to prove the triangles congruent by SAS.

Since the triangles are congruent, all the corresponding parts of the triangles are congruent.

Multiple Choice

What congruency postulate proves the triangles congruency?

• SAS

• ASA

• SSS

• HL

Multiple Choice

Are the triangles congruent, if yes, why?

• SSS

• ASA

• HL

• Not Congruent

Multiple Choice

∆ABC≅∆XYZ

which is true?

• AB≅XY

• BC≅CB

• AB≅YX

• AB≅XZ

Multiple Select

After proving the triangles are congruent by HL Theorem, which statements are true for CPCTC?

• $$\angle X\cong\angle V$$

• $$\angle U\cong\angle X$$

• $$XW\cong VW$$

• $$\angle XUW\cong\angle UWV$$

Multiple Select

After proving the triangles are congruent by SAS postulate, which statements are true for CPCTC?

• $$\angle D\cong\angle A$$  

• $$\angle B\cong\angle D$$  

• $$DE\cong AE$$  

• $$EC\cong AB$$  

• $$BA\cong DC$$  

Multiple Choice

If ∆PIG ≅ ∆COW, WO≅?

• GI

• PI

• PG

Multiple Choice

If ΔDEF ≅ ΔRST, then we can conclude that DE = what?

• ST

• RS

• DF

• RT

When do I use CPCTC?

​In a Geometric Proof.

If you are trying to prove two corresponding parts of a triangle are congruent, and they aren't a reflexive side, vertical angles, alternate interior angles, then CPCTC is the way to go.

In the figure shown,

PROVE: ∡A≅∡E

FIRST--prove the triangles congruent.

THEN once the triangles are congruent, all corresponding parts are ≅

CPCTC is the last REASON

​Example:

In the figure shown,

PROVE: ∡A≅∡E

• Before using CPCTC, show that the two triangles are congruent.

​REMEMBER:

Multiple Choice

Determine the reasons for the remaining steps of the proof.

• Definition of Isosceles Triangle; HL; CPCTC

• Definition of Isosceles Triangle; SAS; CPCTC

• Isosceles Triangle Theorem; HL; CPCTC

• Isosceles Triangle Theorem; SAS; CPCTC

Multiple Choice

Determine the reasons for the remaining steps of the proof.

• Definition of Midpoint; Converse of Isosceles Triangle Theorem; SAS; CPCTC

• Definition of Midpoint; Isosceles Triangle Theorem; HL; CPCTC

• Isosceles Triangle Theorem; Converse of Isosceles Triangle Theorem; HL; CPCTC

• Isosceles Triangle Theorem; Converse of Isosceles Triangle Theorem; SAS; CPCTC

Multiple Choice

Determine the reasons for the remaining steps of the proof.

• Alternate Interior Angles Theorem; Vertical Angles Theorem; AAS; CPCTC

• Definition of Parallel Lines; Vertical Angles Theorem; AAS; CPCTC

• Alternate Interior Angles Theorem; Reflexive Property; ASA; CPCTC

• Definition of Parallel Lines; Reflexive Property; ASA; CPCTC