Triangle Proofs in General
Date: 11/19/2001 at 20:49:48 From: Ashley Subject: Triangle proofs Hello, I'm having a difficult time with proofs in general. I am able to do the really simple proofs but as they get harder and longer I get really lost, especially with the method cpctc after you prove triangles congruent. I am able to start the proof but am never able to complete it. I will show you an example of one I am having difficulty with: C / \ / \ given: AB is a straight line, ca~=cb,<1~=<2 /1 /\2 \ prove: <3~=<4 / / \ \ / /3 4\ \ A-D-----E---B- I know that the first step has to be the given. But then I don't know where to go from there. Should I prove the first triangle congruent (tri. acd~=tri. bcd)? And if I do, how do I get the third side congruent. Because we already have a side and an angle, therefore is it SAS, because we need the other side? If you can help and understand what I am asking, that would be greatly appreciated. Thanks so much. Ashley
Date: 11/19/2001 at 23:40:55 From: Doctor Peterson Subject: Re: Triangle proofs Hi, Ashley. We have a FAQ on proofs that contains the sort of thing I am about to tell you, and more: http://mathforum.org/dr.math/faq/faq.proof.html A lot of people write to tell us they get stuck as soon as they have written down the "givens." I think that's something like "writer's block"; let's call it "prover's block." What do you do when you sit down in front of a blank sheet of paper, or a pair of empty columns? Let's imagine another situation in which you have no idea what to do: you've just parachuted from a disabled plane and find yourself in unknown country, not knowing just where you are or how to get back to civilization. What would you do? I would probably first climb a tree or otherwise get the best view I can of the territory. In your case, you might start by drawing the picture and marking the known facts, as you have; then look it over and familiarize yourself with it. Think not only about what you are told, but about what you see, since too often textbooks don't do a good job of stating everything in the "givens," and force you to make some assumptions from the picture, such as that D and E are on segment AB, and the points are in the order ADEB. Look for triangles that MIGHT be congruent; you can't just assume that, and the picture might be deliberately drawn to hide such things (if you didn't draw your own picture, which is always a good idea!), but it helps to have an idea what you might be able to prove true. In this example, ACD and BCE are a good choice. Here's my personal copy of the picture, on which if I had room I would mark the known pairs of congruent sides or angles: C + // \\ /1/ \2\ / /3 4\ \ +---+-------+---+ A D E B I see from the fact that AC=BC that triangle ACB is isosceles, and that carries with it some extra information, such as that the base angles are equal. That's the sort of thing you want to run through your mind as you look around the problem. Next I would decide where I want to go, and choose a possible route. I might know that there is a town near the lake, and therefore decide to find a stream and follow it downstream. I wouldn't expect to know exactly where to go without a map, but I would want to have some general ideas of the direction I'm heading. In your proof, you want angles 3 and 4 to be equal; what "lake" is that fact on the "shore" of? It sounds like the same fact I mentioned before: the base angles of an isosceles triangle. So if I can prove that DCE is an isosceles triangle, I will be able to get to the goal. What I've really done so far is to think from both ends of the problem: where can I go from where I am (the givens), and how might I get to the goal? Now it's time to start trudging along. I'll never get anywhere unless I start moving; even if I don't know just where to go, I may run into something (like a stream running downhill) that I can follow. You might take a pair of triangles that look congruent and make a list of their "corresponding parts" to see if you have enough to prove congruence. (Don't assume it has to be SAS; there's always ASA.) You can use both given facts and implications that you noticed as you looked around. One way to help yourself get started is to avoid writing anything down in the official columns yet; you're just scouting around, and don't have to get anything in the right order yet. Just jot down any ideas you have. If you like writing (or if you don't), you may have some favorite methods of organizing thoughts for an essay; use any method you want to keep track of your ideas. Eventually, something will click, and you'll realize that you have what you need to get at least a good part of the way toward the goal. Write that down in a half-orderly way so you won't forget it, and then look around again. I've avoided being too precise here to encourage you to go at it yourself; but there should be enough here to help you move forward. If you need more help with this problem, please write back and show me how far you are able to get, even if it's just disorganized ideas. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 11/20/2001 at 20:41:23 From: Ashley Subject: Re: Triangle proofs Thank you so much for the advice. I went back and tried this problem after I wrote you and I did something similar to what you were saying. I usually like to look at the picture I drew and think of the things in it like if there are vertical angles, supplementary angles, complementary angles etc., and see how I can go from there. I got a lot further than I had when I wrote you and I think that I did pretty well.
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