Proving Trapezoid Congruency

Date: 12/19/2001 at 21:22:33
From: Anonymous
Subject: Geometry (Proving isosceles trapezoid)

I have to prove that the sides of a trapezoid are congruent if the 
diagonals of the trapezoid are congruent. 

I started by stating the bases are parallel to each other and I got 
the alternate interior angles congruent. Then I drew a parallel line 
to one of the diagonals because of the parallel postulate. That was as 
far as I could get. I don't understand what else to do. Please help

-Peter

Date: 12/19/2001 at 22:41:17
From: Doctor Peterson
Subject: Re: Geometry (Proving isosceles trapezoid)

Hi, Peter.

I would look for a pair of congruent triangles with which we can prove 
the final result. If the bases are AD and BC, then for example 
triangles ABD and DCA look as if they should be congruent, and these 
include the diagonals. There's another pair of corresponding sides, 
but I don't immediately see that the included angles have to be 
congruent. How can we show that angles ADB and DAC are congruent?

One way I see is to draw the altitudes from B and C to side AD. Since 
AD and BC are parallel, those are equal. Can you see a way to use this 
fact to prove that the angles are congruent?

What I did here was to work backward: the last steps will involve 
proving two triangles congruent; what do I need to do that, and how 
can I get there? Another thing that led me to this idea was imagining 
what it would take to construct a trapezoid with congruent diagonals. 
I might start with parallel lines on which the bases have to lie, and 
two equal sticks for the diagonals. When I saw that in order to go 
from one line to the other they would have to be at the same angle 
(and why), that gave me the idea I needed. 

This is how proofs are generally found: working from both ends and 
playing with the materials we have in order to see what we can do with 
them. It's fun that way, too!

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