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/
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