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### Proving Lines Congruent

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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
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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/
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Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Triangles and Other Polygons

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