Congruent Triangles in a Rectangle

Date: 11/11/1999 at 15:15:28
From: Lacey
Subject: Congruent Triangles (Proof)

Dr. Math -

I've been working on this problem for my Geometry class for days now. 
I'm not very good at proofs, so I am struggling to even get started. 
Here is the information:

     Given: AB is parallel to RT
            AR is perpendicular to AB
            BT is perpendicular to RT
            AB is congruent to RT
            AR is congruent to TB
     Prove: Triangle ABR is congruent to Triangle TRB

(The picture is a rectangle with a diagonal line from corner R to 
corner B going up. Point A is in the upper left hand corner, R is 
bottom left, B is upper right, and T is bottom right.  T and A equal 
90 degrees.)

I hope you have enough information. I am currently trying to get the 
various postulates straight from each other; the Side by Side 
Postulate, Side-Angle-Side Postulate, and the Angle-Side-Angle 
Postulate. Any ideas on how I can remember them?

Thanks a bunch!
Lacey

Date: 11/11/1999 at 17:11:56
From: Doctor Peterson
Subject: Re: Congruent Triangles (Proof)

Hi, Lacey.

You gave me wonderful information! You described the picture 
perfectly, which shows me you have what it takes to prove congruence, 
because you know how to describe something accurately enough so I can 
draw it for myself. That's exactly what a congruence proof is all 
about.

It sounds like what you really need is to get familiar with SSS, SAS, 
and ASA so you can recognize them and know how to use them. Let's 
focus on that rather than on this particular problem.

First, take a look at the first part of this answer from our archives:

  Theorems for Quadrilaterals
  http://mathforum.org/dr.math/problems/bethmarie.11.12.99.html   

It describes the three congruence theorems.

You can remember which ways work by just picturing how you can 
construct them, or by just remembering you have to have either all the 
sides (SSS), or one thing and the two on either side of it (SAS or 
ASA). To put it another way, you always need three things, and either 
one, two, or three of them can be sides - call them the one-stick, 
two-stick, and three-stick methods of building a triangle. Just make 
sure the angle is in the right place if you have only one of them.

Now how do you apply this? One way is to redraw (or re-imagine) the 
two triangles in the same position, so you can more easily recognize 
what goes together.

Here's your original problem:

     A           B
      +---------+
      |       / |
      |     /   |
      |   /     |
      | /       |
      +---------+
     R           T

Here are the two triangles pulled apart:

     A          B    B
      +---------+   +
      |       /   / |
      |     /   /   |
      |   /   /     |
      | /   /       |
      +   +---------+
     R    R         T

And here I've turned ABR around so they're facing the same way:

                R              B
                +              +
              / |            / |
            /   |          /   |
          /     |        /     |
        /       |      /       |
      +---------+    +---------+
     B          A   R          T

Notice that the letters of ABR and TRB match: A and T are in the same 
places, and so on. They were named in this order to show which pairs 
of vertices go together.

Your "givens" are actually far more than you need to prove the 
conclusion. You could use any of SSS, SAS, or ASA to do it.

Let's list the parts of the two triangles and what you know about 
them:

     ABR   TRB
     ---   ---
Sides
     AB  = TR (and parallel, too)
     BR  = RB (in fact, they're the same segment)
     RA  = BT
Angles
     RAB = BTR (both right angles)
     ABR = TRB (alternate interior angles)
     BRA   RBT (no immediate relationship)

There are not 3, but 5 things you can see immediately or very easily 
to be the same. See if you can find three that prove congruence. 
(Hint: SSS is probably the easiest.)

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