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### Two-Column Proof of Congruence

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Date: 05/16/2000 at 21:06:29
From: Claire
Subject: Geometry 9 "proofs"

I have no idea whatsoever what I'm doing. It's so overwhelming I can't
example question:

Complete each of the following proofs.

(There's a picture of two triangles put together to make a slanty
rectangle and each corner of this slanty rectangle is a letter. Top
left is G, bottom left is T, bottom right is A, and top right is O.)

Then beside the picture is:

Given: GO || TA,
GT || OA

Show: (triangle sign) GOT = (triangle sign) ATO

Underneath the picture and all that is:

Statement               Reason
GO || TA, GT || OA
<GOT = <ATO
OT = TO
<GTO = <AOT
tri. GOT = tri. ATO
```

```
Date: 05/16/2000 at 22:45:32
From: Doctor Peterson
Subject: Re: Geometry 9 "proofs"

Hi, Claire.

You'll want to look through our Dr. Math FAQ on proofs to get a feel
for what proofs are all about:

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

A proof is just an orderly explanation of why you can be sure
something is true. We take one step at a time and give a reason for
everything we say, so there can be no doubt. In your problem, you are
given the proof (the "statements"), and just have to figure out why
each step was done (filling in the "reasons"). Let's go through it
together. First, I'll draw the picture:

G       O
+-------+
/     / /
/   /   /
/ /     /
+-------+
T       A

Now we'll take the statements one at a time:

>   1. GO || TA, GT || OA

This is easy: it's just what they gave us to start with; so we write
our reason as:

Given

>   2. <GOT = <ATO

Why should these two angles be congruent? Look at where they are in
the figure:

G       O
+-------+
/   2 / /
/   /   /
/ / 2   /
+-------+
T       A

I've marked them with a "2"; this should remind you of a picture you
have seen that looked like this:

------------+
2 /
/
/ 2
+------------

We have two parallel lines and a transversal; the angles "2" are
called alternate interior angles. There's a theorem that "alternate
interior angles are congruent." There's our reason:

alternate interior angles are congruent

This is just a short way of saying "TO is a transversal of the
parallel lines GO and TA (which we know to be parallel by statement
1), so by this theorem, we know that the alternate interior angles GOT
and ATO are congruent."

>   3. OT = TO

Here we have the opposite problem: this is so obvious we wonder why it
was stated at all, and why it will help us. It's here just so a later
statement can refer to it; all it's saying is that OT is equal to
itself. I'm not sure what the standard phrase is for this (after high
school we don't bother with the details of this sort of two-column
proof), but the reason can be something like:

anything is congruent to itself

or

reflexive property of equality

(which is the fancy way of saying the same thing: "reflexive" means
that the equals sign is like a mirror, and the image it "reflects" is
the same as the original.) You probably have a list of names of
obvious rules like this for use in filling in this sort of reason.

>   4. <GTO = <AOT

I'll let you see what the reason for this is; it's essentially the
same as step 2.

>   5. tri. GOT = tri. ATO

Now, how can we prove that two triangles are congruent? You probably
have several theorems along those lines; if we look at the two
triangles, and line them up so the letters match, we can see what
statements 2 through 4 tell us about them:

G       O      A       T
+-------+      +-------+
/   2 /        /   2 /
/   /          /   /
/4/   3        /4/   3
+              +
T              O

I've marked the facts stated in statements 2 through 4 with their
numbers. (You can see why the letters in each of those statements were
in the order they are, such as OT = TO; that makes them match up just
right in this pair of triangles. Everything has been leading up to
this.) Notice that we have two angles and the included side -- ASA.
Now we know why statement 3 was there! And the reason for this final
statement is:

ASA congruence theorem

We're done. Of course, we could actually go further; knowing that
these triangles are congruent, we could continue and prove that pairs
of opposite sides of a parallelogram are congruent. But we were told
to stop here, so we will.

I've included a lot of essential hints in what I've said, such as
drawing the two triangles separately to help you see the parts in the
proof, so take your time thinking through how I did this. Feel free to
write back if you have more trouble with this; it does take time to
get used to the idea of proofs.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry

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