Naming Corresponding Parts of Congruent Figures

Date: 04/23/2003 at 22:56:54
From:  John
Subject: Corresponding parts of congruent figures

Given triangle OPS congruent to triangle TQR, name the corresponding 
sides and angles of the two triangles.

Date: 04/24/2003 at 08:49:14
From: Doctor Peterson
Subject: Re: Corresponding parts of congruent figures

Hi, John.

When we name triangles that are congruent, we name them both in the 
order in which we consider parts to correspond. Your triangles might 
look like this:

          P                       Q
         /  \                    /  \
        /     \                 /     \
       /        \              /        \
      /           \           /           \
     /              \        /              \
    O-----------------S     T-----------------R

So among the vertices, O corresponds to T (in the same position at 
lower left when they are lined up this way), and their angles SOP and 
RTQ are the same. Among the edges, side OS corresponds to TR (on the 
bottom in both), and they have the same length. You just have to list 
the other two pairs of corresponding sides and the other two pairs of 
corresponding angles.

Note that congruent triangles don't HAVE to be facing the same 
direction; they might look like this instead:

          P                 T-----------------R
         /  \                \              /
        /     \               \           /
       /        \              \        /
      /           \             \     /
     /              \            \  /
    O-----------------S           Q

Although TR is no longer on the bottom, it still corresponds to OS 
because it is the side with the same length. You might picture tracing 
OPS on a sheet of paper, labeling the vertices, and cutting it out. 
Then slide it around so that the vertices O, P, and S lie on top of T, 
Q, and R, respectively, in the other triangle, to show that they are 
congruent. (In this case, you will also have to flip the paper over.) 
Whatever "part" of TQR lies under any part of OPS is its corresponding 
part.

Here is a discussion from the Dr. Math archives of how these ideas 
apply in a proof:

   Congruent Parts Congruent Triangles Congruent (CPCTC)
   http://mathforum.org/library/drmath/view/55397.html 

and here is an example where it is used:

   Writing a Proof
   http://mathforum.org/library/drmath/view/55312.html 

If you have any further questions, feel free to write back.

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