Two-Column Proof of Congruence
Date: 05/16/2000 at 21:06:29 From: Claire Subject: Geometry 9 "proofs" I have no idea whatsoever what I'm doing. It's so overwhelming I can't even begin to try and figure it out. Please help me. I'll give you an example question: Complete each of the following proofs. (There's a picture of two triangles put together to make a slanty rectangle and each corner of this slanty rectangle is a letter. Top left is G, bottom left is T, bottom right is A, and top right is O.) Then beside the picture is: Given: GO || TA, GT || OA Show: (triangle sign) GOT = (triangle sign) ATO Underneath the picture and all that is: Statement Reason GO || TA, GT || OA <GOT = <ATO OT = TO <GTO = <AOT tri. GOT = tri. ATO
Date: 05/16/2000 at 22:45:32 From: Doctor Peterson Subject: Re: Geometry 9 "proofs" Hi, Claire. You'll want to look through our Dr. Math FAQ on proofs to get a feel for what proofs are all about: http://mathforum.org/dr.math/faq/faq.proof.html A proof is just an orderly explanation of why you can be sure something is true. We take one step at a time and give a reason for everything we say, so there can be no doubt. In your problem, you are given the proof (the "statements"), and just have to figure out why each step was done (filling in the "reasons"). Let's go through it together. First, I'll draw the picture: G O +-------+ / / / / / / / / / +-------+ T A Now we'll take the statements one at a time: > 1. GO || TA, GT || OA This is easy: it's just what they gave us to start with; so we write our reason as: Given > 2. <GOT = <ATO Why should these two angles be congruent? Look at where they are in the figure: G O +-------+ / 2 / / / / / / / 2 / +-------+ T A I've marked them with a "2"; this should remind you of a picture you have seen that looked like this: ------------+ 2 / / / 2 +------------ We have two parallel lines and a transversal; the angles "2" are called alternate interior angles. There's a theorem that "alternate interior angles are congruent." There's our reason: alternate interior angles are congruent This is just a short way of saying "TO is a transversal of the parallel lines GO and TA (which we know to be parallel by statement 1), so by this theorem, we know that the alternate interior angles GOT and ATO are congruent." > 3. OT = TO Here we have the opposite problem: this is so obvious we wonder why it was stated at all, and why it will help us. It's here just so a later statement can refer to it; all it's saying is that OT is equal to itself. I'm not sure what the standard phrase is for this (after high school we don't bother with the details of this sort of two-column proof), but the reason can be something like: anything is congruent to itself or reflexive property of equality (which is the fancy way of saying the same thing: "reflexive" means that the equals sign is like a mirror, and the image it "reflects" is the same as the original.) You probably have a list of names of obvious rules like this for use in filling in this sort of reason. > 4. <GTO = <AOT I'll let you see what the reason for this is; it's essentially the same as step 2. > 5. tri. GOT = tri. ATO Now, how can we prove that two triangles are congruent? You probably have several theorems along those lines; if we look at the two triangles, and line them up so the letters match, we can see what statements 2 through 4 tell us about them: G O A T +-------+ +-------+ / 2 / / 2 / / / / / /4/ 3 /4/ 3 + + T O I've marked the facts stated in statements 2 through 4 with their numbers. (You can see why the letters in each of those statements were in the order they are, such as OT = TO; that makes them match up just right in this pair of triangles. Everything has been leading up to this.) Notice that we have two angles and the included side -- ASA. Now we know why statement 3 was there! And the reason for this final statement is: ASA congruence theorem We're done. Of course, we could actually go further; knowing that these triangles are congruent, we could continue and prove that pairs of opposite sides of a parallelogram are congruent. But we were told to stop here, so we will. I've included a lot of essential hints in what I've said, such as drawing the two triangles separately to help you see the parts in the proof, so take your time thinking through how I did this. Feel free to write back if you have more trouble with this; it does take time to get used to the idea of proofs. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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