Thinking About ProofsDate: 09/24/97 at 15:29:27 From: Erika Subject: PROOFS How do you know what statement to write next when you're doing a proof? And what are the reasons that you use? ex. given: m<4 + m<6 = 180 prove: m<5 = m<6 K / \ / \ / \ / \ ____4_/5______6_\ J L Date: 09/24/97 at 18:03:11 From: Doctor Pete Subject: Re: PROOFS Hi, Probably the most difficult part of proving something is where to start. It takes a lot of practice, and trial and error. Since you are more interested in the thought process than the solution, let me tell you what went through my mind as I solved your example. "Well, I don't need to know anything about the lengths of the triangle, because all I've been given are angles. I remember that the sum of the angles of a triangle is always 180 degrees, but that doesn't help because I don't know what angle 5 is nor do I know what angle JKL is. Well, I know that angle 4 + angle 6 = 180. Ah! But I also know that angle 4 + 5 = 180, because together they form the straight line which contains JL. So if angle 4 + angle 6 = 180 = angle 4 + angle 5, then angle 4 + angle 6 = angle 4 + angle 5, and once we subtract angle 4 from both sides of this equation, we get our result, angle 6 = angle 5." Notice that I tried to rule out some possible ways of proving the theorem, based on what I was given (and importantly, what I was not given). I then relied on previous theorems or things I already knew that were relevant. I applied what I was given in the problem to this previous knowledge, and I was able to determine that some of my ideas didn't work. Luckily, I recognized that I could apply one of my ideas to the solution, and at that point I was able to see that this was the right direction to take. The rest was a matter of finishing the details. While this is not always how proofs are made, it gives you an idea of the process, which is more or less "educated trial-and-error." You guess, you eliminate possibilities, narrow it down, and push the line of reasoning further. Some proofs, like this one, only required one "breakthrough" idea, namely, the fact that supplementary angles add to 180 degrees. However, some proofs require applying many ideas, all in the right order. That's why they're difficult, because at each point in the proof there are many directions to take. Knowing which is the right one is something that takes practice and patience. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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