Indirect Proof of Parallel Lines

Date: 11/26/2001 at 18:12:52
From: Ryan O'Donnell
Subject: Indirect proof of parallel lines

Dr. Math,

I have asked my high school geometry class to prove indirectly that 
parallel lines have the same slope. Unfortunately, I cannot figure out 
how to do it myself!

I assumed that "2 parallel lines DO NOT have the same slope" and then 
tried to draw two interseting lines and show that this violates the 
statement that they are parallel. Unfortunately, saying 'parallel 
lines' is equivalent to saying 'non-intersecting lines'. This reduces 
my assumption to "2 non-intersecting lines DO NOT have the same slope" 
or "2 non-intersecting lines have to intersect."

Do you have any suggestions on how to complete this proof? Currently 
I have been wondering whether or not there is a way to curve one of 
the lines in order to make them intersect and observe that this 
contradicts the definition of line. What do you think?

Thanks for the help.
Ryan O'Donnell

Date: 11/27/2001 at 18:07:13
From: Doctor Achilles
Subject: Re: Indirect proof of parallel lines

Hello Ryan,

Thanks for writing to Dr. Math.

The logical statement you are trying to prove is:

  If two lines are parallel, then 
  they will have the same slope.

One way to do this is to assume the existence of a counter-example.  
So:

  There are two lines that are parallel 
  that do not have the same slope.

Unfortunately, it's impossible to draw this counter-example, because 
(as you've quite rightly argued) it is a self-contradictory idea. The 
argument you gave above is a sound logical argument why no counter-
example can exist. You could be satisfied with that if you were just 
trying to prove this to a bunch of people who were already familiar 
with geometric proofs, but fortunately there is a way to make a visual 
proof as well.

In order to get there, you have to do just a bit of introductory 
logic. For any sentence of the form:

  if P then Q

you can write three other related sentences.  The converse:

  if not P then not Q

The inverse:

  if Q then P

And the contrapositive:

  if not Q then not P

The converse and inverse are not terribly interesting for our 
purposes, but remember that the contrapositive is logically equivalent 
to the original sentence.

For more information and a good discussion that's relevant to your 
question, check out:

  Logic of Indirect Proofs
  http://mathforum.org/dr.math/problems/carnrike.10.16.96.html   

Going over the idea of contrapositives with your class may take a day 
or so, but it's something they're going to do with geometry anyway.

So you can prove:

  If two lines are parallel, then they will have the same slope.

By proving:

  If two lines have a different slope, then they will not be parallel.

Then you can draw two lines with a different slope and show that they 
are not parallel. This way you can get at the proof in a visual way.

I hope this is helpful. If you have more questions or would like to 
talk about how to go over with this proof in class some more, please 
write back.

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