Two-Column Proof About KitesDate: 11/09/1999 at 22:15:30 From: Amy DeMent Subject: Two-Column Proofs I am in geometry, and I am completely lost on proofs. I understand you have some givens, and you have to prove something, but as to the steps in between, I am clueless. If I have one side of the proof I can get the other side, or if I am looking at a completed proof I can see how it was done, but I don't understand how to come up with the statements. For instance, here is an example that I copied down from the board that I don't understand. The only reason it is completed is because it was done for me on the board. Given: AP = BP AQ = BQ Prove: PQ is perpendicular to AB P /| \ / | \ A /__R___\B \ | / \ | / \| / Q PROOF Statements Reasons ---------- ------- 1. AP = BP 1. Given 2. AQ = BQ 2. Given 3. PQ = PQ 3. Reflexive property 4. triangle PAQ = triangle PBQ 4. SSS 5. <APR = <BPR 5. CPCTC* 6. PR = PR 6. Reflexive property 7. triangle PRA = triangle PRB 7. SAS 8. m<PRA = m<PRB 8. CPCTC* 9. <PRA, <PRB are linear pair 9. If alt. int. lines are congruent, the lines are parallel 10. m<PRA = m<PRB = 90 10. If 2 <'s are congruent and form a linear pair, each is a right angle 11. PQ is perpendicular to AB 11. Def. of perpendicular lines *CPCTC = Corresponding parts of congruent triangles are congruent And that is the whole proof. Could you please explain why each step is done and how you came up with it? Amy Date: 11/10/1999 at 09:17:26 From: Doctor Peterson Subject: Re: Two-Column Proofs Hi, Amy. Thanks for a well-written and thorough question. I'd recommend looking through our FAQ on proofs: http://mathforum.org/dr.math/faq/faq.proof.html This is a compilation of previous answers to questions, and will show you both examples of how we've done simple proofs, and explanations of ways to think about it. I'll deal with your particular example, but there you will find several different ways to approach proofs that will be useful to you. > Given: AP = BP > AQ = BQ > > Prove: PQ is perpendicular to AB > > P > /| \ > / | \ > A /__R___\B > \ | / > \ | / > \| / > Q > > PROOF > Statements Reasons > ---------- ------- > 1. AP = BP 1. Given > 2. AQ = BQ 2. Given Having written down the givens, and without having looked at the rest of the proof yet, let's think about what we have and where we want to get to - look over the territory before we start marching across it. We have a kite shape, and want to prove that the "sticks" are perpendicular. My first thought is that they're held that way because the string around the edge of the kite pulls equally in both directions and holds them straight. That has nothing to do with geometry, but maybe it can give me a feel for how this will work. Then I look at which edges are equal, and realize that we have two isosceles triangles, APB and AQB. PQ, if the conclusion is right, will be an altitude of both triangles. Next, looking ahead to our goal, I look for some possible congruent triangles. APB and AQB are not going to be congruent, but within each of them the altitude ought to form two congruent triangles, APR = BPR and AQR = BQR. If I can prove these, maybe I'll be going in the right direction. Now let's see what comes next... > 3. PQ = PQ 3. Reflexive property > 4. triangle PAQ = triangle PBQ 4. SSS Ah! They thought of something better: each side of the whole kite will be congruent. This is essentially because the kite is symmetrical: the sides are the same because opposite sides match. We stated PQ = PQ just to get all three sides so we can apply the SSS theorem. Now where will this get us? It says nothing yet about the angles at O; yet it seems related to the goal because if we fold the kite over along line PQ, the fact that A and B will meet says the line between them is perpendicular to the fold (or "mirror"). How do we show that? > 5. <APR = <BPR 5. CPCTC* > 6. PR = PR 6. Reflexive property > 7. triangle PRA = triangle PRB 7. SAS Just what I was thinking: the congruence of PAQ and PBQ, which we proved on the basis of their sides, tells us something about their angles, which we can use in other triangles. Here we've shown that the top two triangles are congruent. Remember how I said the sticks in a kite are held perpendicular because they're pulled equally from each side? That's what we're seeing now: angles ARP and BRP will be congruent because the two sides are the same, and two congruent angles that are supplementary have to be right angles. Let's see if that's how they finish this. > 8. m<PRA = m<PRB 8. CPCTC* > 9. <PRA, <PRB are linear pair 9. If alt. int. lines are > congruent, the lines are > parallel > 10. m<PRA = m<PRB = 90 10. If 2 <'s are congruent and > form a linear pair, each > is a right angle > 11. PQ is perpendicular to AB 11. Def. of perpendicular lines Yep, I was right. So we're done. Let's look back over what we did. If I were doing this on my own, rather than trying to follow (and sometimes anticipate) someone else's proof, I would have thought something like this: I'm given some equal sides; from that I may be able to prove some triangles congruent. At the other end, I want to come out with a statement about angles, and since right angles are congruent to one another, that can probably come from some congruent triangles. I'd work from both ends: the triangles I can prove congruent from the givens are PAQ and PBQ; the triangles I'd want to prove congruent in order to get to my conclusion are either PRA and PRB or QRA and QRB (I could do the proof using either pair.) Then I'd look for a connection between PAQ and PRA, and would see that they share an angle and a side. At that point I'd be almost done. As I explain in some of my answers in the FAQ, this illustrates that there can be a lot of looking around before we hit on the actual path to a proof, and that a proof can be likened to building a bridge, starting from both shores and meeting in the middle. There may also be some wrong turns along the way, or some good ideas that aren't needed for this particular proof - but that might give you ideas for another. Also, you don't have to think of the statements one at a time. They come in groups: prove these triangles congruent, then prove those triangles congruent. Get the main idea, then work out how to say the details. I hope this helps you out. Learning proofs can be a challenge, because it's a new idea to most students, and a new way of thinking; but if you can use your experience in other kinds of thinking, you can do fine. Let me know if you need any more help. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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