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Two-Column Proof of Congruence


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 
even begin to try and figure it out. Please help me. I'll give you an 
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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