Isosceles Trapezoid ProofDate: 01/23/2002 at 17:04:03 From: Ana Subject: Understanding Theorems and 2 column proofs The proof on my latest quiz was as follows: Given: ABCD is an isosceles trapezoid with bases BC and AD Prove: ABCD is an isosceles trapzoid ^ / B _1________ C / \ / \ A--------------D The quiz I took gave the conclusions and we had to find the justifications. Conclusions My Justifications 1) AD//BC 1) defn. of trapezoid 2) m<1=m<A 2) defn. of corr. <'s 3) m<D=m<A 3) subsition (from given, step 2 and 3) 4) ABCD is isos. trap. 4) defn. of isos. trap. From my justifications I got step 1 and 4 correct. The answers the teacher gave us for stepa 2 and 3 were: 2) If // lines then corr. <'s equal. 3) trans. prop. from given, step 2, and step 3. Could you please explain to me what I did wrong in solving this proof? Ana Date: 01/23/2002 at 22:43:08 From: Doctor Peterson Subject: Re: Understanding Theorems and 2 column proofs Hi, Ana. I'm confused about the problem itself; did it really say the same thing as both given and conclusion? That doesn't make a lot of sense, so I hope you copied it wrong! The problem with your reason for step 2 is that it is not the DEFINITION you are really using. The definition of corresponding angles is something like "angles in a pair of figures that correspond." (I can't think offhand how I would actually define it.) What you are using is a THEOREM about corresponding angles in a figure consisting of a pair of parallel lines and a transversal. That is what the teacher's answer is saying; probably you could look in your book and find a theorem number or title for it, but a brief statement of the theorem is fine. Since you seem to have misstated what was given, I'm not fully sure what step 3's reason refers to; is it given that angles 1 and D are congruent? Assuming it is something like that, your answer isn't too far off here. The transitive property looks a lot like substitution, and I wouldn't really count it wrong to say the latter when the former is true. In a sense the transitive property is a special case of substitution; it says that if a = b and b = c, then a = c and you could express this in terms of substitution as by substituting c for b in "a = b", we find that a = c. But since the transitive property is stronger (more clearly defined, perhaps), it is better to use that whenever you can, and to reserve "substitution" for more complex cases where you substitute within an expression, rather than replacing a whole side of an equality. But again, that's far from a major error. The basic idea is right either way. The distinction between definition and theorem is a lot more important. It's mostly a matter of paying close attention to how your book states such things; learn the definitions and the theorems. (I would keep my own list, if the book doesn't collect them in one place for you.) It can be tricky sometimes, because one book's definition can be another book's theorem. This happens when an entity has several properties, and you have to choose which to use as the defining property, and which to prove. There's an example right here: I would define an isosceles trapezoid as one whose legs are congruent, since the word "isosceles" is Greek for "equal legs." Your book apparently defines it as having congruent base angles, which I would prefer to prove from my definition. This is a case where I can't tell, without seeing your book, whether you are right in calling the latter a definition rather than a theorem. But keep track of the definitions used in your own class, and you should have no trouble. For more about proofs you might read some of the answers in our FAQ on proofs: http://mathforum.org/dr.math/faq/faq.proof.html We talk about how to look at a proof in general, how to find your way from the givens to the conclusion, and how to write it up. In general, a proof is like a persuasive essay: you don't write it all at once, but gather ideas, find the line of reasoning you like, and then put it in order for the final draft. So you should not expect to go straight through a proof. Just play around with the ideas until you find the path. There are more specific ideas in the FAQ. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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