Writing a Proof
Date: 05/16/2001 at 21:31:23 From: Tony Ventura Subject: Solving proofs I really don't understand how or what to do when doing proofs. Is there a certain way I should go about doing them? If so, what? Here is one of the problems I had to do. __ ~ __ 1. Given: BO = Co; __ ~ __ AO = DO ~ Prove: (angle) <B = <C Proof: __________Statements______________________Reasons____________________ | 1. __ ~ __ __ ~ __ | 1. BO = CO; AO = DO | | ~ | 2. <AOB = <DOC | 2. | ~ | 3. Triangle ABO = Triangle DCO | 3. | ~ | 4. <B = <C | 4. | __ KEY: BO means segment BO ~ = means congruent < means angle
Date: 05/16/2001 at 23:04:03 From: Doctor Peterson Subject: Re: Solving proofs Hi, Tony. First, if you haven't visited our FAQ, go there; we have some good discussions of proofs: http://mathforum.org/dr.math/faq/faq.proof.html Now let's look at your proof. I'll simplify the notation a bit, so we both know the correct symbols: Given: BO = CO AO = DO Prove: <B = <C Statements Reasons 1. BO = CO; AO = DO 1. 2. <AOB = <DOC 2. 3. Triangle ABO = Triangle DCO 3. 4. <B = <C 4. First we can draw a picture; it seems that some facts were left to be deduced from the picture, rather than being explicitly stated as "givens," which is a poor practice. I'll have to make some assumptions: A + / \ / \ / \ B +-------+-------+ C O \ / \ / \ / + D I'm assuming (based on the proof) that "angle B" is ABO and "angle C" is DCO, and that AOD and BOC are collinear. These should be "given", and not assumed from the picture - take a few points off for the textbook authors! Now back to you. The first statement has an easy reason: that's what you were told is true. Write "given." The whole proof just supposes that we know these things, and tells us what we can then know to be true. The second is based on something you know about the two angles named, namely that they are "vertical angles." Write a brief statement of what you know about vertical angles. The third statement has to be based on a theorem about congruent triangles, since that's what it says. What theorems do you know with which you can prove that two triangles are congruent? Look at the facts you have about the two triangles; it can help to list the corresponding parts of ABO and DCO: ABO DCO --- --- sides: AB DC BO CO AO DO angles: ABO DCO BOA COD OAB ODC The previous statements in the proof tell you that certain of these parts are congruent. Mark those on the chart by putting an equals sign between them; you may want to write the number of the statement that says each is true. Then look to see if what you know fits any theorems. You'll find that it does; later you'll have to look for additional facts you can prove that will fill in the missing parts, but here they're making it easy by giving you all the steps. The last step is an automatic result of the triangles being congruent; it is often stated as "corresponding parts of congruent triangles are congruent." If ABO and DCO are congruent triangles, then angles ABO and DCO (and any other pairs) must match. Now, in this problem, you didn't really have to do the proof; you are acting as an artist's apprentice, just filling in between the lines already drawn. As you learn the art of proof (and it really is an art, sometimes very beautiful, but also tedious at times), you will get more used to the process and be able to sketch out the shape of the proof yourself first, and let someone else fill in the details. (That's what mathematicians do in practice - we usually write just enough to show other mathematicians that it works, and let them fill in the details we all know how to do.) The fun part is the discovery; but you have to start somewhere! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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