Congruent Parts Congruent Triangles Congruent (CPCTC)

Date: 11/28/2001 at 17:50:12
From: Brittany
Subject: Congruent Parts Congruent Triangles Congruent 

Dear Dr. Math,

I am a freshman and my teacher taught us CPCTC, but I do not 
understand when to use it, why we use it, and how it proves anything. 
Can you help?

Thanks,
Brittany

Date: 11/28/2001 at 23:21:45
From: Doctor Peterson
Subject: Re: Congruent Parts Congruent Triangles Congruent 

Hi, Brittany.

CPCTC is a complicated way to say something obvious: if two things are 
identical, then every part of them is identical too. For example, if I 
wrote you a letter and made a Xerox copy of it, then the first word on 
both copies would be the same. If I copied a photograph of myself, 
then the left ear on both copies would look the same.

In a proof, you often want to show that two line segments are the same 
length. If you can use theorems to show that two triangles of which 
these segments are part are congruent, then since every part of the 
two triangles is the same, these sides must be the same too.

The trick is to make sure you choose CORRESPONDING parts! If you 
compared two pictures of me, but claimed that the left ear of one had 
to be the same as the right ear of the other, you would be wrong. One 
ear may have been injured or have an earring in it, and the other not. 
You can only say they must be the same if they are both the same ear 
on identical pictures. The same is true with triangles. That's why, 
when we talk about congruence, we have to name the triangles in the 
same order. These triangles are congruent:

          B             E
          +             +
        /  \          /  \
      /     \       /     \
    +--------+    +--------+
   A          C  D          F

I have to say that ABC is congruent to DEF, to show the order in which 
I am taking the triangles; then sides AB and DE are congruent, and so 
are BC and EF. I can't say that triangles ABC and EFD are congruent, 
because AB is not the same length as EF. But when I state clearly 
which parts correspond, then I can say that AB and DE are 
corresponding parts, and therefore must be congruent. If they weren't, 
the triangles wouldn't be the same!

When do you use CPCTC? When you know that two triangles are congruent, 
and want to say something about their angles or sides.

Why? In order to prove congruence of angles or segments.

I hope that helps. If you have more questions, feel free to write 
back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/