CPCTC Proofs

Videos, worksheets, solutions, and activities to help Geometry students learn about CPCTC. CPCTC is an acronym for Corresponding Parts of Congruent Triangles are Congruent.

What is CPCTC?
CPCTC is an acronym for corresponding parts of congruent triangles are congruent. CPCTC is commonly used at or near the end of a proof which asks the student to show that two angles or two sides are congruent. It means that once two triangles are proven to be congruent, then the three pairs of sides that correspond must be congruent and the three pairs of angles that correspond must be congruent.

How do we use CPCTC to prove corresponding parts of triangles?
CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent and is used to justify two triangles are congruent in a proof.
Examples:

1. A and B are the endpoints of a bridge going over a river. What is the length of the bridge?
2. Given YW bisects XZ. XY ≅ YZ. Prove that ∠XYW ≅ ∠ZYW.
3. Given D(-5, -5), E(-3, -1), F(-2, -3), G(-2, 1), H(0, 5), and I(1, 3). Prove that ∠DEF ≅ ∠GHI.

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Using CPCTC - Corresponding Parts of Congruent Triangles are Congruent
Anytime you are required to prove corresponding parts of congruent triangles congruent, you will be doing a triangle proof. The two examples in this post use AAS and SAS before proving the other part of the triangle congruent using CPCTC.
Examples:

1. Given SL ≅ SR, ∠1 ≅ ∠2. Prove ∠3 ≅ ∠4.
   Now that we have proved the triangles congruent and angle 3 and angle 4 are congruent using CPCTC, what other congruence statements can you make from the diagram?
2. Given ∠Q ≅ ∠R, ∠QPS ≅ ∠RSP. Prove SQ ≅ PR

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SSS, SAS, ASA proofs with CPCTC
Congruent triangle are triangles in which corresponding angles and sides are congruent.

Example 1:
If triangle CAT ≅ triangle DOG, list all of the congruencies.
How many congruencies does it take for us to show that the two triangles are congruent?
How many additional congruencies could we deduce?

Example 2:
Given RZ bisects ∠TRS, ∠3 ≅ ∠4. Prove ∠S ≅ ∠T

Example 3:
Given AB bisects CD, ∠C ≅ ∠D. Prove ∠A ≅ ∠B

Example 4:
Given M is the midpoint of AB, ∠1 ≅ ∠2, ∠3 ≅ ∠4. Prove AC ≅ BD.

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How to identify and use Corresponding Parts of Congruent Figures?
Learn to identify and apply the CPCTC rule
Example:
Use the congruent triangles below and answer the following questions.
(a) What side of ABC corresponds to side YZ?
(b) Name the side of triangle XYZ that corresponds to side AB.
(c) What is the measure of side ZX?
(d) Find the perimeter of triangle XYZ?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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