Parallel Lines: Two Column Proof

Date: 09/09/98 at 23:06:05
From: Turtle
Subject: Geometry

Hello, 

I was wondering if there is any way that you could break down the steps 
in doing a two-column proof? One that we had to do for homework is:

    Given: Angle 1 congruent angle 2, angle 3 congruent angle 4
    Prove: n parallel p

               l         m
             /        3/
      ------/---------/------n
           /         /
        1 /        2/
     ----/---------/------p    angle 5 is under angle 2
        /4        /

Date: 09/10/98 at 17:04:39
From: Doctor Peterson
Subject: Re: Geometry

Hi, Turtle. 

Two-column proofs are a little foreign to most of us - even to 
mathematicians, who don't usually use such a rigid way of writing a 
proof once they have learned what it means to prove something. The idea 
is to force you to think very clearly and express yourself very 
precisely. Unfortunately, no one really thinks that way, so if you're 
just shown a two-column proof without an explanation of how someone 
produced it, it seems like either magic ("how did he do that?") or a 
waste of time ("why did he bother to do all that?").

I suggest that you first try to prove your goal without thinking about 
the details of the two columns. People too often get bogged down in 
the details ("what is the exact reason for this step?" "is this 
legal?") when the important thing about a proof is to learn the logic 
behind it.

A proof is sort of a bridge from the "mainland" of known truth to an 
"island" you want to get to. In your case, you have been given a 
platform you are supposed to start from (the "givens"), and you have 
some set of definitions, postulates, and already-proven theorems that 
you can use. Think of those as materials you can use to build the 
bridge:

    Given: angle 1 congruent angle 2
           angle 3 congruent angle 4
    Prove: n parallel p

               l         m
             /        3/
      ------/---------/------n
           /         /
        1 /        2/
     ----/---------/------p    
        /4        /

Now you need to take a quick helicopter ride over the territory between 
the starting and ending points and see what you can recognize as useful 
stepping stones. It will take some practice to get used to what sorts 
of things you should look for. Just make a list of facts you either can 
deduce from the givens or can use to get to the goal. This is like 
building a bridge by starting at both ends and working toward the 
middle:

    Givens:
        Angle 1 = angle 2
        Angle 3 = angle 4

    Deductions (from the givens):
        Angle 1 = angle 4 (vertical angles)
        Angle 1 = angle 2 implies that l and m are parallel

    Possibilities (for proving the goal):
        n is parallel to p if angles 2 and 3 are equal

    Aha! I found a link.
        I have 2 = 1 and 1 = 4 and 4 = 3, so I can prove that 2 = 3.
        (Notice I don't need all of my deductions)

Now you have the idea of a proof, and you can start working out the 
details. We can locate the stepping stones our bridge will use:

    Given:
        Angle 1 = angle 2
        Angle 3 = angle 4
    Steps:
        Angle 1 = angle 4 (vertical angles)
        2 = 3 (because 2 = 1 = 4 = 3)
        n is parallel to p (because 2 = 3)

It's starting to look like a proof. But now we need to organize it by 
stating clearly just what the reason is for each step - laying boards 
between the stepping stones - and filling in some steps that help 
clarify what we are doing:

    Given:
        Angle 1 congruent angle 2
        Angle 3 congruent angle 4

    Statement:                     | Reason:
    -------------------------------+--------------------------
     1: Angle 1 congruent angle 4  | vertical angles are equal
     2: Angle 1 congruent angle 2  | given
     3: Angle 2 congruent angle 4  | both equal to angle 1
     4: Angle 3 congruent angle 4  | given
     5: Angle 2 congruent angle 3  | both equal to angle 4
     6: Line n is parallel to p    | transversal makes equal angles

The exact wording will depend on what your text gives as the names of 
the theorems you use, how your teacher asks you to lay out the proof, 
and so on. That isn't the important thing, and you should not worry 
about the "rivets" in your proof (as long as they pass the teacher's 
inspection). Think about it; you've just built a bridge to new 
territory. When you get further, you'll have much more significant 
proofs to write. None of them will be a Golden Gate, but some will be 
pretty impressive. Right now, you're just hopping a puddle, and it 
might not look like much, but it's good practice for building the big 
bridges.

Here's a nice brief explanation of this process in our Dr. Math 
archives, if you'd like another perspective:

   http://mathforum.org/dr.math/problems/2_col_proofs.html   

Another answer in the archives:

   http://mathforum.org/dr.math/problems/crystal9.12.98.html   

I hope this helps you learn to write proofs.

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