Proving Lines Congruent

Date: 03/29/2002 at 14:01:50
From: Allison
Subject: Geometry Proofs

Dr. Math,

I am having a little trouble understanting proofs in geometry. One in 
particular that I am having trouble with is as follows:

I hope you can understand the drawing.

The picture is of a slanted rectangle with A and B on the bottom; A 
and D is the left side, and B and C is the right side, and D and C is 
the top.  Line DB cuts the rectangle diagonally in half.  L and M are 
on line DB.  BCM forms an isosceles triangle and ADL forms another 
isosceles triangle. 

You are given that angle DAL is congruent to angle BCM
                   angle CDL is congruent to angle ABM
                   line DC is congruent to line BA

You have to prove line AL is congruent to line CM

I would greatly appreciate your help in solving this problem.

Thank you,
Allison

Date: 03/29/2002 at 16:49:57
From: Doctor Peterson
Subject: Re: Geometry Proofs

Hi, Allison.

It does help a lot to have someone to talk to about proofs; they are 
often hard for students to grasp at first. Let's work through this one 
together.

I presume that by "slanted rectangle" you mean that it looks like a 
parallelogram; and that nothing tells you explicitly that this is 
true, or that what look like isosceles triangles really are - these 
are just descriptions of what the picture looks like, for my guidance. 
Let me know if that is wrong.

Here is my version of the picture:

              D                z                 C
               +--------------------------------+
              /  \  y                        / /
             /     \                       / x/
            /        \  L               /    /
           /           +             /      /
          /         /    \         / ?     /
         /     ?  /        \    /         /
        /      /             +           /
       /    /              M   \        /
      /x  /                      \     /
     / /                        y  \  /
    +--------------------------------+
   A                 z                B

I've marked "x" and "y" where pairs of angles are congruent, and z for 
the congruent sides. You want to prove that the two segments marked 
with "?" are congruent.

I would start by looking at the implications of the facts you know, so 
we can get a feel for what I have to work with:

   The x's suggest that triangles DAL and BCM may be congruent, though
   we don't know enough yet to say they are.

   The y's on opposite sides of transversal BD tell us that AB and DC
   are parallel - that will likely be one of the steps in the proof.

   That and the z's tell me that this is in fact a parallelogram; you
   probably have a theorem or two that will prove that. Other facts  
   arise from this, such as that AD and BC are congruent and parallel, 
   and that opposite angles are congruent. These are all worth paying 
   attention to!

Now we can look from the other end: what would help in proving our 
goal? Well, if, as I suggested, we can prove that triangles DAL and 
BCM are congruent, that will get us to the goal. What would it take to 
prove that? There are several triangle congruence theorems, each of 
which requires knowing three parts of the triangles. What parts do we 
know?

    angle DAL = BCM  - given
    side DA = BC  - because we have a parallelogram

There's another pair of angles we can prove congruent without much 
effort, using the facts we now have to work with.

See if you can put these ideas together into a proof.

If you haven't seen our FAQ yet, it has some good general ideas on how 
to approach proofs:

    http://mathforum.org/dr.math/faq/faq.proof.html   

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