Isosceles Trapezoid Proof

Date: 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/