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## 4.4 CPCTC Using Corresponding Parts of Congruent Triangles are Congruent

Rooted in the definition of congruent triangles, the acronym CPCTC stands for Corresponding Part of Congruent Triangles are Congruent. Basically, you have to prove the triangles congruent with what is given and once the triangle congruence is established the remaining corresponding parts are congruent.

Anytime you are required to prove corresponding parts of congruent triangles congruent, you will be doing a triangle proof, meaning that you will need to prove the triangles congruent with ASA, AAS, SAS, SSS, or HL. The examples are solved with AAS and SAS to prove the other part of the triangle congruent using CPCTC.

## Example 1 Using CPCTC to Prove Two Segments Congruent

It is given that angle BAE is congruent to angles BCD and that segment AE is congruent to segment CD. With the given marked on the diagram you should recognize that there is one pair of corresponding angles and one pair of corresponding segments congruent. That leaves you with establishing ASA, AAS, or SAS in the two triangles.

Notice, triangles BAE and BCD share the vertex B. By the reflexive property of congruence, angle B is congruent to itself. Now triangles BAE and BCD are congruent because of AAS.

Since segment AB and CB are corresponding part of two triangles, they are congruent by CPCTC.

## Example 2 Using CPCTC to Prove Two Segments Congruent

I have not yet used this in my blog, it is a modified two column proof. There is still the statements and reasons, but the reasons are included in a set of parenthesis. The statements are numbered starting with 1.

1.            segment DA is congruent to segment DC (Given)

2.            angle BDA is congruent to angle BDC (Given)

3.            segment BD is congruent to segment BD (reflexive property of congruence)

4.            triangle ABD is congruent to triangle CBD (SAS)

5.            angle ABD is congruent to angle CBD (CPCTC)

As you can see, using CPCTC is just a matter of completing a triangle congruence proof first.

Until next time,

Mr. Pi

### 3 Responses

1. Mr. Pi,

I hope you don’t mind, I’ll just test latex if it works on comments.

$\overline{DA} \cong \overline{DC}$
$\\angle{BDA} \cong \angle {BDC}$

Guillermo

2. sorry, i think my code is wrong on the second line. anyway, at least we know it’s working.

• Guillermo,

I think you missed your slash. I hope I code it right $\angle {BDA} \cong \angle {BDC}$

Mr. Pi