The Order of a Proof

Date: 01/29/99 at 08:14:32
From: Stan D.
Subject: Geometry

Dear Dr. Math,

My question is about proofs. My Geometry book only describes the two-
column proof twice, and it doesn't give too many details. I cannot 
figure out if their statements and reasons are completely random in 
their ordering (other than the "given" and the "to prove" which are 
always first and last), or if there is a particular method for the 
order in which they should be placed.

Figuring out which theorems, postulates, and definitions to use in a 
proof are no problem. However, they seem to just pull statements out of 
thin air, not taking the time to describe how even in the study guide. 

Any help that you could give me would be greatly appreciated and I 
thank you for your time.

Date: 01/29/99 at 12:14:09
From: Doctor Rob
Subject: Re: Geometry

Thanks for writing to Ask Dr. Math!

The only requirement for ordering the steps and reasons is that if
Step A depends on Step B, then A should follow B. That is why the 
given is first and the conclusion is last, and the same logic applies
to all the intermediate steps.

For example, prove: In a triangle ABC, if side CA = side CB, then 
<A = <B.

Proof:

Steps:                           Reasons:
1. CA = CB.                      1. Given.
2. CB = CA.                      2. Step 1 and Symmetric Law of 
                                    Equality.
3. <C = <C.                      3. Reflexive Law of Equality.
4. triangle BCA =~ triangle ACB  4. Steps 1, 2, 3, and S.A.S. theorem.
5. <A = <B.                      5. Step 4 and corresponding parts of
                                    congruent figures are congruent.
Q.E.D.

In this proof, Step 2 depends on Step 1, so must follow it. Step 4 
depends on Steps 1, 2, and 3, so must follow them. Step 5 depends on 
Step 4, so must follow it. The only freedom you have is the placement 
of Step 3, which can go anywhere before Step 4 (and then you would 
renumber the steps and reasons to be in order 1 through 5).

Here is a diagram of the structure of the proof:

   1 ==> 2 ==> 4
   1 ========> 4 ==> 5
         3 ==> 4

The hard part is not figuring the order of the steps, it is figuring 
what steps are good ones to take! Usually the mathematician does this 
kind of thing backwards (yes, backwards!). He says, "If I could prove 
that triangles BCA and ACB are congruent, then I could conclude that 
<A = <B in one step. Furthermore, it seems from my diagram that this 
is probably true." So he tries to prove the triangle congruence. "If I 
could find two sides and an included angle in BCA equal to two sides 
and an included angle in ACB, I could use S.A.S. to prove the 
triangles congruent." So he hunts for equal sides and angles. This is 
the way it goes. When done, the proof is presented in the forward 
manner, not providing a hint of the backward process used to generate 
it. Sometimes one can work from the given forwards and from the 
conclusion backwards and meet somewhere in the middle.

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