Writing a Proof

Date: 05/16/2001 at 21:31:23
From: Tony Ventura
Subject: Solving proofs

I really don't understand how or what to do when doing proofs. Is 
there a certain way I should go about doing them? If so, what? 

Here is one of the problems I had to do. 

                 __ ~ __
       1. Given: BO = Co;
                 __ ~ __
                 AO = DO
                            ~
          Prove: (angle) <B = <C

          Proof:

__________Statements______________________Reasons____________________
                                 |
 1. __ ~ __  __ ~ __             |   1.
    BO = CO; AO = DO             |
                                 |
         ~                       |
 2. <AOB = <DOC                  |   2.
                                 |
                 ~               |  
 3. Triangle ABO = Triangle DCO  |   3.
                                 |
       ~                         |
 4. <B = <C                      |   4.
                                 |

     __
KEY: BO   means segment BO

     ~
     =    means congruent

     <    means angle

Date: 05/16/2001 at 23:04:03
From: Doctor Peterson
Subject: Re: Solving proofs

Hi, Tony.

First, if you haven't visited our FAQ, go there; we have some good 
discussions of proofs:

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

Now let's look at your proof. I'll simplify the notation a bit, so we 
both know the correct symbols:

    Given: BO = CO
           AO = DO
    Prove: <B = <C

    Statements                         Reasons
 1. BO = CO; AO = DO                1. 
 2. <AOB = <DOC                     2.
 3. Triangle ABO = Triangle DCO     3.
 4. <B = <C                         4.

First we can draw a picture; it seems that some facts were left to be 
deduced from the picture, rather than being explicitly stated as 
"givens," which is a poor practice. I'll have to make some 
assumptions:

          A
          +
         / \
        /   \
       /     \
    B +-------+-------+ C
             O \     /
                \   /
                 \ /
                  +
                  D

I'm assuming (based on the proof) that "angle B" is ABO and "angle C" 
is DCO, and that AOD and BOC are collinear. These should be "given", 
and not assumed from the picture - take a few points off for the 
textbook authors!

Now back to you.

The first statement has an easy reason: that's what you were told is 
true. Write "given." The whole proof just supposes that we know these 
things, and tells us what we can then know to be true.

The second is based on something you know about the two angles named, 
namely that they are "vertical angles." Write a brief statement of 
what you know about vertical angles.

The third statement has to be based on a theorem about congruent 
triangles, since that's what it says. What theorems do you know with 
which you can prove that two triangles are congruent? Look at the 
facts you have about the two triangles; it can help to list the 
corresponding parts of ABO and DCO:

               ABO   DCO
               ---   ---
    sides:     AB    DC
               BO    CO
               AO    DO

   angles:     ABO   DCO
               BOA   COD
               OAB   ODC

The previous statements in the proof tell you that certain of these 
parts are congruent. Mark those on the chart by putting an equals sign 
between them; you may want to write the number of the statement that 
says each is true. Then look to see if what you know fits any 
theorems. You'll find that it does; later you'll have to look for 
additional facts you can prove that will fill in the missing parts, 
but here they're making it easy by giving you all the steps.

The last step is an automatic result of the triangles being congruent; 
it is often stated as "corresponding parts of congruent triangles are 
congruent." If ABO and DCO are congruent triangles, then angles ABO 
and DCO (and any other pairs) must match.

Now, in this problem, you didn't really have to do the proof; you are 
acting as an artist's apprentice, just filling in between the lines 
already drawn. As you learn the art of proof (and it really is an art, 
sometimes very beautiful, but also tedious at times), you will get 
more used to the process and be able to sketch out the shape of the 
proof yourself first, and let someone else fill in the details. 
(That's what mathematicians do in practice - we usually write just 
enough to show other mathematicians that it works, and let them fill 
in the details we all know how to do.) The fun part is the discovery; 
but you have to start somewhere!

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