Parallel Lines: Two Column ProofDate: 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/ |
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