Geometry ProofsDate: 11/07/2001 at 16:29:00 From: Victoria Nosser Subject: Geometry proofs When my teacher is writing proofs I understand them, but I am having trouble writing them on my own. I can figure out whether the figure is A.S.A. or l.l., etc., but most of the time I have left out a a lot of the statements. How can I just write these things correctly? Thanks, Victoria Date: 11/07/2001 at 17:10:22 From: Doctor Peterson Subject: Re: Geometry proofs Hi, Victoria. I'd be happy to look over a sample of your work and see how we can help you. Just do a proof for me as well as you can, and tell me where you are dissatisfied with it. You may also find some very helpful ideas on proofs in our FAQ: About Proofs http://mathforum.org/dr.math/faq/faq.proof.html A lot of people have asked for help either with the basic concept of proof, how to get a proof started, or how to write it out, and the answers collected there should help you with the same issues. I'll make one broad comment to start you off: to avoid getting too frustrated, try to focus on the overall idea of a proof, rather than the details. If you see what's going on, but you can't say it just right, you've got the important part, so feel good about getting as far as you did. The "big picture" of a proof is like reaching a mountain top and seeing the scenery; writing out the proof is the long hike back to reality. It's important to finish, but take time to enjoy the view first! It can be breathtaking sometimes (though not on your first few problems). - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 11/08/2001 at 23:30:17 From: Victoria Nosser Subject: Re: Geometry proofs Thank you so much for offering up the time to help me out with this, Doctor Peterson. Here is a problem that I really messed up on: GIVEN: In the figure, /\(triangle)ABC is isosceles (AC=BC), and AD=BE. PROVE: (a) <a = <b (b) /\DCA ~=(congruent) /\ECB DIAGRAM: C / /\ \ / = = \ /a/ \b \ --'--------'-- D A B E (This diagram looks bad, but what it is supposed to look like is one big triangle with anisosceles triangle in the middle which is /\ABC. and the <a is the angle at the corner of <DAC. <b is at the corner of <EBC. The two ='s are there to say that AC=BC and the two apostrophes are there to say that AD=BE) Here is what I would put on my proof: Statements Reasons ------------------------------------------------------- 1. /\ABC is an isosceles 1. Given AC=BC and AD=BE 2. <BAC=<ABC 2. Definition of isosceles /\ 3. <a=<b 3. (I know this is wrong but this is all I know to put.) Because it is an isosceles and the legs are always equal, the base is the unequal side. 4. /\DCA ~= /\ECB 4. Because both /\'s are sharing two sides that are equal. Therefore the last side is the same length in both /\'s and results in the /\'s being ~= I know I messed up big time on that. Here is what my mom's answer key says I was supposed to have. STATEMENTS REASONS ------------------------------------------------------- 1. Isosceles /\ABC with 1. Given AC=BC AD=BE 2. <BAC=<ABC 2. Angles opposite equal sides in a /\ are equal. 3. <a + <BAC = a st. angle, 3. The sum of the angles about a point and on one side of a st. line is a st. <b + <ABC = a st. angle angle. 4. <a + <BAC = <b + <ABC 4. substitution axiom 5. <a = <b 5. substitution axiom 6. /\DCA ~= /\ECB 6. S.A.S. See, I understand all of that, I guess I just always find shortcuts in my proofs. What should I do? Victoria Nosser Date: 11/09/2001 at 12:01:29 From: Doctor Peterson Subject: Re: Geometry proofs Hi, Victoria. Thanks for writing back - you've given me just the kind of information I need in order to help out, which few students take the time to do. You even did pretty well drawing and explaining an awkward diagram! Have you ever heard of a "paragraph proof"? The two-column style proof is often taught in American schools as a way to force you to show a proof step by step, giving every detail. Mathematicians don't write proofs that way, and I've seen some who were educated elsewhere express amazement at seeing a two-column proof. Instead, they just write a paragraph explaining their reasoning. They don't need to state every step and reason precisely, because they know other mathematicians know all the basics and can fill in those details. The important thing is to show that there is a complete path from the "givens" to the goal. You have written paragraph proofs as your "reasons"! That's not to say that they are good proofs; you don't have the experience yet to know what is important to say, and what constitutes a sufficient reason for a statement. But on the other hand, I'm not happy with all the book's reasons either; there's no reason your answer has to duplicate theirs, because many of the steps here are "trivial" ones for which it's hard to give a clear reason. What's wrong with your shortcuts is not that they are short (it's great to be able to see the truth of a statement at once) but that they don't clearly communicate to me why it's true, and that's what proof is all about. It's really not much different from writing a persuasive essay; you can't just say "This is true because it feels right to me," but you have to state your evidence and convince a (friendly) skeptic. This is why two-column proofs are taught; by requiring you to break your thinking up into short statements, it forces you to analyze your thinking and explain each step. The hard part is learning how to see the little steps in your thinking, and express them succinctly. Let's take a look at each of your reasons, and see how we can improve them. First, let me have some fun trying to draw your picture: C + // \\ / / \ \ / a/ \b \ +---+-------+---+ D A B E It's still necessary to explain separately what a and b are for clarity, and I don't even try to mark congruent segments on this sort of diagram. In any case, one of my pet peeves about geometry texts is that they often make you depend too much on the picture; a theorem should always state EVERY relevant fact (such as that D, A, B, and E are collinear and in that order) explicitly in words, so you can draw the picture yourself and not have to guess which facts about the picture are to be assumed. So I'd rather have you tell me about a picture in words than draw it perfectly. Now let's start with your proof: 1. /\ABC is an isosceles 1. Given AC=BC and AD=BE There should be nothing to say here, since it's just "givens" and agrees with your book; but I prefer to put each fact on its own line, so I can refer to it clearly later. I would say 1. Tri ABC is iso, 1. Given with AC=BC 2. AD=BE 2. Given Next: 2. <BAC=<ABC 2. Definition of isosceles /\ This is insufficient, because there's a difference between the definition and your statement. You have to be precise about what the definition is; presumably your book defines an isosceles triangle as one with two equal legs. Since that definition says nothing about the angles of the triangle, you can't use it as a reason for such a statement. But your book probably does give a theorem (soon after the definition, perhaps) that says that the base angles of an isosceles triangle are congruent. That had to be proved, but once you have done that, you can use it as a reason. So I would state the theorem briefly as my reason: 3. <BAC=<ABC 3. Base angles of iso.tri. are congruent They've said the same thing, just a little differently. The exact wording doesn't matter; the fact that we refer back to a theorem that they proved for you is the essential part. We must build our building from the "steel" of theorems, not from the "straw" of unsupported jumps, even if they are true. Always check what theorems you have to work from, and use those rather than "common knowledge". 3. <a=<b 3. (I know this is wrong but this is all I know to put.) Because it is an isosceles and the legs are always equal, the base is the unequal side. You've seen that the base angles being congruent is the basis of this statement; but you haven't given enough reasoning to show how that is connected to what you are saying here. What you need to do is to make one statement at a time to demonstrate that connection. Statement 2 said that the base angles are congruent; but angles a and b are not the base angles. What are they, then? Angle BAC is one base angle; angle a is its supplement. What we want to say, then, is that the supplements of congruent angles are congruent. I would be satisfied with that as your reason here; it's a clear statement of why you think a and b are congruent. The book has apparently not proved such a particular fact, and instead takes it in steps. There are different ways to prove this; here's a way using measures of angles and algebraic facts: 4. m<BAC = 180 - m<a 4. Supplementary angles 5. m<ABC = 180 - m<b 5. Supplementary angles 6. 180 - m<a = 180 - m<b 6. Transitivity of equality 7. m<a = m<b 7. Subtract 180 from each side and negate Notice that I'm not being picky about the exact terms to use for each little fact; I did use the term "transitivity," which is just a short way to say that things equal to equals are equal, but I don't care how precisely you say that 6 implies 7. The important thing is that it's a short enough step that anyone can see that 7 follows from 4 and 5. 4. /\DCA ~= /\ECB 4. Because both /\'s are sharing two sides that are equal. Therefore the last side is the same length in both /\'s and results in the /\'s being ~= This is a very broad representation of the Side-Angle-Side theorem. Unfortunately, you didn't actually mention the angle, which is of course a central (pun intended) part of this theorem. I think the book was too brief; I like to state not just what theorem was applied, but also how it was applied. I would say this: 8. tri DCA = tri ECB 8. SAS, from 1, 7, 2 This way I'm saying just which sides and angle I'm using, which makes it clearer for the reader. You don't need to write a paragraph telling what this means to you and why it ought to be true; the theorem does all the heavy work, and you just have to point to it. Now let's back away from this and look at what you can do to improve your proofs in general. On the one hand, I don't want you getting bogged down in details; the important thing is to get moving in geometry and see the main ideas. Never let yourself be held back by something little like trying to decide whether you used the substitution property or the transitive property! But on the other hand, your main difficulty seems to be that you don't pay enough attention to the details that ARE important. I'd like you to focus on the concept of theorems. Each reason you give should be based on a specific theorem (or postulate or definition) - a known fact you can rely on, rather than a general sense that something is true. In this case, there are two main theorems you should have seen and mentioned in your proof, the one about base angles and the SAS congruence theorem. I don't care about the other things, like transitivity; those are really just names given to "obvious" facts so we can state a reason. (Of course, in a careful development of math they would have to be proved from axioms; but at this level they can just be taken as known.) The important thing is that your reasons are not meant to be vague descriptions of your thinking, but precise statments of a single proven fact. I like to see what's behind your thinking (which is one reason I prefer paragraph proofs); but you need to anchor everything in theorems. I would recommend making a list of all the definitions, postulates, theorems, and so on in your book as you come to them. Then you can look through the list to find a reason for each statement; if you don't find one, you know you have to break down your thinking into smaller steps. I hope this helps; I'd like you to practice this a bit and then send in another of your proofs, one that you think comes closer to the goal without being a copy of what's in the book, and we can work on developing your own style of proof. One of the interesting things about proofs is that they are so much more like an English class than the math you've seen before! There's a lot of room for individual style, and a lot of creativity involved. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 11/09/2001 at 16:30:43 From: Victoria Nosser Subject: Re: Geometry proofs Thank you so much. I am going to be looking over this and working on some more proofs. I will write you back soon to let you know how well I am doing. I really appreciate your breaking down of what I am doing in my proofs. :) I'll be sure to refer any one I know to ya'lls Web site if they have trouble. Talk to you soon. Thanks, Victoria |
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