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### Isosceles Trapezoid Proof

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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
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.

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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Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Triangles and Other Polygons

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