How to Build a Proof

Date: 05/18/99 at 18:37:19
From: Veronica
Subject: Geometry

In my geometry class we are doing proofs and I just don't get them. I 
have the first step, which is the given, but I don't know where to go 
from there. Can you please help me with the problem?

Given: Triangle ABC is a right triangle, with a right angle at 3.
Prove: Angle A and angle B are complementary angles.

Statement                                  Reason
1) Triangle ABC is a right                 1) Given
   triangle with right angle 3.

Thank you,
Veronica

Date: 05/19/99 at 12:16:08
From: Doctor Peterson
Subject: Re: Geometry

Hi, Veronica.

Proofs are probably something pretty new to you, and it does take time 
to get a feel for what makes a proof good enough and how you can find 
the way to prove something. It's really more like writing an essay 
than like the math you've done before now - more creative and less 
mechanical. That makes it harder, but also more rewarding and even 
fun.

One thing that's important is not to sit staring at an empty two-
column chart. Our goal is to make a proof, not to fill in two columns; 
if we think about the columns too early it can keep us from the goal.

I like to think of a proof as a bridge, or maybe a path through a 
forest: you have to start with some facts you are given, and find a 
way to your destination. You have to start out by looking over the 
territory, getting a feel for where you are and where you have to go -
what direction you have to head, what landmarks you might find on the 
way, how you'll know when you're getting close.

In this case, what we start with is a right triangle; I suspect you 
were given a picture that shows that angle 3 is at vertex C, because 
that really should be stated as one of the "givens." Let's draw a 
picture to we see what we have:

         A
         +
        /|
       / |
      /  |
     /  3|
    +----+
    B    C

We have built the structure on one shore of the "river" we want to 
cross:

    Statement                       Reason
    ---------                       ------
    Triangle ABC is a right         Given
      triangle with right angle 3

We also know where we want to end up:

    Angle A and angle B are         ?
      complementary angles

Let's look around a bit. What does "complementary" mean? We want to 
show that A + B = 90 degrees. That tells us we want to work with the 
angles of this triangle. What do we know about angles of a triangle? 
You may have several theorems to consider; one that should come to 
mind quickly is that the sum of the angles is 180 degrees.

What I've been doing here is looking at the tools and materials I have 
to build our bridge or path. So far I know I start with a triangle, 
one of whose angles is known; I want to get an equation involving the 
other two angles; and I have a theorem about all the angles of a 
triangle. That sounds promising!

Let's lay out what we have as the beginning of a proof:

    Statement                       Reason
    ---------                       ------
    Triangle ABC is a right         Given
      triangle with right angle 3
     .
     .
     .
    A + B + C = 180                 Sum of angles theorem
     .
     .
     .
    Angle A and angle B are         ?
      complementary angles

Now what do we need to fill in the gaps? Well, let's rewrite the 
other statements in a way that looks more like the theorem we hope we 
can use:

    Statement                       Reason
    ---------                       ------
    Triangle ABC is a right         Given
      triangle with right angle 3
    C = 90                          Definition of right angle
     .
     .
     .
    A + B + C = 180                 Sum of angles theorem
     .
     .
     .
    A + B = 90                      ?
    Angle A and angle B are         Definition of complementary
      complementary angles

Do you see how we're working both forward and backwards? That's where 
my bridge-building analogy comes in: you can work on both ends of a 
bridge and let them meet in the middle.

Okay, how can we show that A + B = 90? Since C is 90, we can just do 
some algebra, subtracting the equation C = 90 from A + B + C = 180. 
You've done the same sort of thing in algebra without having to write 
it as a two-column proof; here we have to be able to say briefly why 
this works, and you may have been given a list of basic facts about 
algebra that you can use as reasons. I'll just call it "Subtracting 
equals from equals."

Now all that's left is to put it together into a coherent proof. That 
means we have to figure out how to state each step clearly; each step 
has to follow from steps that have already been written; and each step 
has to be small enough that we can give the reason without any huge 
leaps that would be hard to explain. Let's try:

    Statement                       Reason
    ---------                       ------
    Triangle ABC is a right         Given
      triangle with right angle 3
    C = 90                          Definition of right angle
    A + B + C = 180                 Sum of angles theorem
    A + B = 90                      Subtracting equals from equals
    Angle A and angle B are         Definition of complementary
      complementary angles

This could use some rewriting to make it clearer, perhaps, and you 
should use the correct symbolism for "the measure of angle C" rather 
than just say "C"; but it does the job. A lot of students worry 
whether they have stated their reasons well enough, and usually most 
of the worries are about the most trivial steps - the facts we may not 
even be able to give a name to because they're obvious. That's why 
texts often spend a lot of time giving special names to obvious facts, 
like "Corresponding Parts of Congruent Triangles are Equal" - not 
because they're really important, but because the two-column format 
requires you to say something, and they don't want you to agonize over 
the wording. Real mathematicians don't worry about such details, as 
long as they know each step is true. The important thing for you is 
that you found a path. That's something to celebrate - don't they have 
a little party when they finish a bridge, and nail a tree to the top 
or something? Maybe you can draw a little tree on your proof, and take 
a break before making the final copy.

It took a while to get this done, but it's rewarding to have explored 
a jungle like this (or is it a river?) and find that we know our way 
around well enough to build a path where there wasn't one before, just 
as it's rewarding to have written a paper that explained thoughts we 
didn't know we had, or facts we had to research. Or to explain to 
someone how to do something they didn't know how to do. I hope I've 
done that; feel free to write back for more help, because this is a 
New Idea that takes getting used to.

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