Geometry Proofs

Date: 11/07/2001 at 16:29:00
From: Victoria Nosser
Subject: Geometry proofs

When my teacher is writing proofs I understand them, but I am having 
trouble writing them on my own. I can figure out whether the figure is 
A.S.A. or l.l., etc., but most of the time I have left out a a lot of 
the statements. How can I just write these things correctly? 

Thanks,
Victoria

Date: 11/07/2001 at 17:10:22
From: Doctor Peterson
Subject: Re: Geometry proofs

Hi, Victoria.

I'd be happy to look over a sample of your work and see how we can 
help you. Just do a proof for me as well as you can, and tell me where 
you are dissatisfied with it.

You may also find some very helpful ideas on proofs in our FAQ:

   About Proofs
   http://mathforum.org/dr.math/faq/faq.proof.html   

A lot of people have asked for help either with the basic concept of 
proof, how to get a proof started, or how to write it out, and the 
answers collected there should help you with the same issues.

I'll make one broad comment to start you off: to avoid getting too 
frustrated, try to focus on the overall idea of a proof, rather than 
the details. If you see what's going on, but you can't say it just 
right, you've got the important part, so feel good about getting as 
far as you did. The "big picture" of a proof is like reaching a 
mountain top and seeing the scenery; writing out the proof is the long 
hike back to reality. It's important to finish, but take time to enjoy 
the view first! It can be breathtaking sometimes (though not on your 
first few problems).

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 11/08/2001 at 23:30:17
From: Victoria Nosser
Subject: Re: Geometry proofs

Thank you so much for offering up the time to help me out with this, 
Doctor Peterson. 

Here is a problem that I really messed up on:

GIVEN: In the figure, /\(triangle)ABC is isosceles (AC=BC), and AD=BE.
PROVE: (a) <a = <b
       (b) /\DCA ~=(congruent) /\ECB


DIAGRAM:                  C  
                        / /\ \
                       / = = \
                      /a/   \b \
                    --'--------'--
                    D   A    B   E

(This diagram looks bad, but what it  is supposed to look like is one 
big triangle with anisosceles triangle in the middle which is /\ABC. 
and the <a is the angle at the corner of <DAC. <b is at the corner of 
<EBC. The two ='s are there to say that AC=BC and the two apostrophes 
are there to say that AD=BE)

Here is what I would put on my proof:

Statements                 Reasons
-------------------------------------------------------
1. /\ABC is an isosceles   1. Given
   AC=BC and AD=BE
2. <BAC=<ABC               2. Definition of isosceles /\
3. <a=<b                   3. (I know this is wrong but this is all  
                              I know to put.) Because it is an 
                              isosceles and the legs are always equal,
                              the base is the unequal side.
4. /\DCA ~= /\ECB          4. Because both /\'s are sharing two sides 
                              that are equal. Therefore the last side 
                              is the same length in both /\'s and
                              results in the /\'s being ~=


I know I messed up big time on that. Here is what my mom's answer key 
says I was supposed to have.

STATEMENTS                                REASONS
-------------------------------------------------------
1. Isosceles /\ABC with        1. Given
   AC=BC AD=BE
2. <BAC=<ABC                   2. Angles opposite equal sides in a /\ 
                                  are equal.

3. <a + <BAC = a st. angle,    3. The sum of the  angles about a point   
    and                           on one side of a st. line is a st. 
   <b + <ABC = a st. angle        angle.
                                     
4. <a + <BAC = <b + <ABC       4. substitution axiom
                                       
5. <a = <b                     5. substitution axiom
                                       
6. /\DCA ~= /\ECB              6. S.A.S.

See, I understand all of that, I guess I just always find shortcuts in 
my proofs. What should I do? 

Victoria Nosser

Date: 11/09/2001 at 12:01:29
From: Doctor Peterson
Subject: Re: Geometry proofs

Hi, Victoria.

Thanks for writing back - you've given me just the kind of information 
I need in order to help out, which few students take the time to do. 
You even did pretty well drawing and explaining an awkward diagram!

Have you ever heard of a "paragraph proof"? The two-column style proof 
is often taught in American schools as a way to force you to show a 
proof step by step, giving every detail. Mathematicians don't write 
proofs that way, and I've seen some who were educated elsewhere 
express amazement at seeing a two-column proof. Instead, they just 
write a paragraph explaining their reasoning. They don't need to state 
every step and reason precisely, because they know other 
mathematicians know all the basics and can fill in those details. The 
important thing is to show that there is a complete path from the 
"givens" to the goal.

You have written paragraph proofs as your "reasons"!

That's not to say that they are good proofs; you don't have the 
experience yet to know what is important to say, and what constitutes 
a sufficient reason for a statement. But on the other hand, I'm not 
happy with all the book's reasons either; there's no reason your 
answer has to duplicate theirs, because many of the steps here are 
"trivial" ones for which it's hard to give a clear reason. What's 
wrong with your shortcuts is not that they are short (it's great to be 
able to see the truth of a statement at once) but that they don't 
clearly communicate to me why it's true, and that's what proof is all 
about. 

It's really not much different from writing a persuasive essay; you 
can't just say "This is true because it feels right to me," but you 
have to state your evidence and convince a (friendly) skeptic. This is 
why two-column proofs are taught; by requiring you to break your 
thinking up into short statements, it forces you to analyze your 
thinking and explain each step. The hard part is learning how to see 
the little steps in your thinking, and express them succinctly.

Let's take a look at each of your reasons, and see how we can improve 
them. First, let me have some fun trying to draw your picture:

            C
            +
          // \\
        / /   \ \
      / a/     \b \
    +---+-------+---+
   D    A       B    E

It's still necessary to explain separately what a and b are for 
clarity, and I don't even try to mark congruent segments on this sort 
of diagram. In any case, one of my pet peeves about geometry texts is 
that they often make you depend too much on the picture; a theorem 
should always state EVERY relevant fact (such as that D, A, B, and E 
are collinear and in that order) explicitly in words, so you can draw 
the picture yourself and not have to guess which facts about the 
picture are to be assumed. So I'd rather have you tell me about a 
picture in words than draw it perfectly.

Now let's start with your proof:

1. /\ABC is an isosceles   1. Given
   AC=BC and AD=BE

There should be nothing to say here, since it's just "givens" and 
agrees with your book; but I prefer to put each fact on its own line, 
so I can refer to it clearly later. I would say

1. Tri ABC is iso,       1. Given
   with AC=BC   
2. AD=BE                 2. Given

Next:

2. <BAC=<ABC               2. Definition of isosceles /\

This is insufficient, because there's a difference between the 
definition and your statement. You have to be precise about what the 
definition is; presumably your book defines an isosceles triangle as 
one with two equal legs. Since that definition says nothing about the 
angles of the triangle, you can't use it as a reason for such a 
statement. But your book probably does give a theorem (soon after the 
definition, perhaps) that says that the base angles of an isosceles 
triangle are congruent. That had to be proved, but once you have done 
that, you can use it as a reason. So I would state the theorem briefly 
as my reason:

3. <BAC=<ABC              3. Base angles of iso.tri. are congruent

They've said the same thing, just a little differently. The exact 
wording doesn't matter; the fact that we refer back to a theorem that 
they proved for you is the essential part. We must build our building 
from the "steel" of theorems, not from the "straw" of unsupported 
jumps, even if they are true. Always check what theorems you have to 
work from, and use those rather than "common knowledge".

3. <a=<b                   3. (I know this is wrong but this is all  
                              I know to put.) Because it is an 
                              isosceles and the legs are always equal,
                              the base is the unequal side.

You've seen that the base angles being congruent is the basis of this 
statement; but you haven't given enough reasoning to show how that is 
connected to what you are saying here. What you need to do is to make 
one statement at a time to demonstrate that connection.

Statement 2 said that the base angles are congruent; but angles a and 
b are not the base angles. What are they, then? Angle BAC is one base 
angle; angle a is its supplement. What we want to say, then, is that 
the supplements of congruent angles are congruent. I would be 
satisfied with that as your reason here; it's a clear statement of why 
you think a and b are congruent. The book has apparently not proved 
such a particular fact, and instead takes it in steps. There are 
different ways to prove this; here's a way using measures of angles 
and algebraic facts:

4. m<BAC = 180 - m<a       4. Supplementary angles
5. m<ABC = 180 - m<b       5. Supplementary angles
6. 180 - m<a = 180 - m<b   6. Transitivity of equality
7. m<a = m<b               7. Subtract 180 from each side and negate

Notice that I'm not being picky about the exact terms to use for each 
little fact; I did use the term "transitivity," which is just a short 
way to say that things equal to equals are equal, but I don't care how 
precisely you say that 6 implies 7. The important thing is that it's a 
short enough step that anyone can see that 7 follows from 4 and 5.

4. /\DCA ~= /\ECB          4. Because both /\'s are sharing two sides 
                              that are equal. Therefore the last side 
                              is the same length in both /\'s and
                              results in the /\'s being ~=

This is a very broad representation of the Side-Angle-Side theorem. 
Unfortunately, you didn't actually mention the angle, which is of 
course a central (pun intended) part of this theorem. I think the book 
was too brief; I like to state not just what theorem was applied, but 
also how it was applied. I would say this:

8. tri DCA = tri ECB       8. SAS, from 1, 7, 2

This way I'm saying just which sides and angle I'm using, which makes 
it clearer for the reader. You don't need to write a paragraph telling 
what this means to you and why it ought to be true; the theorem does 
all the heavy work, and you just have to point to it.

Now let's back away from this and look at what you can do to improve 
your proofs in general. On the one hand, I don't want you getting 
bogged down in details; the important thing is to get moving in 
geometry and see the main ideas. Never let yourself be held back by 
something little like trying to decide whether you used the 
substitution property or the transitive property! But on the other 
hand, your main difficulty seems to be that you don't pay enough 
attention to the details that ARE important.

I'd like you to focus on the concept of theorems. Each reason you give 
should be based on a specific theorem (or postulate or definition) - a 
known fact you can rely on, rather than a general sense that something 
is true. In this case, there are two main theorems you should have 
seen and mentioned in your proof, the one about base angles and the 
SAS congruence theorem. I don't care about the other things, like 
transitivity; those are really just names given to "obvious" facts so 
we can state a reason. (Of course, in a careful development of math 
they would have to be proved from axioms; but at this level they can 
just be taken as known.) The important thing is that your reasons are 
not meant to be vague descriptions of your thinking, but precise 
statments of a single proven fact. I like to see what's behind your 
thinking (which is one reason I prefer paragraph proofs); but you need 
to anchor everything in theorems.

I would recommend making a list of all the definitions, postulates, 
theorems, and so on in your book as you come to them. Then you can 
look through the list to find a reason for each statement; if you 
don't find one, you know you have to break down your thinking into 
smaller steps.

I hope this helps; I'd like you to practice this a bit and then send 
in another of your proofs, one that you think comes closer to the goal 
without being a copy of what's in the book, and we can work on 
developing your own style of proof. One of the interesting things 
about proofs is that they are so much more like an English class than 
the math you've seen before! There's a lot of room for individual 
style, and a lot of creativity involved.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 11/09/2001 at 16:30:43
From: Victoria Nosser
Subject: Re: Geometry proofs

Thank you so much. I am going to be looking over this and working on 
some more proofs. I will write you back soon to let you know how well 
I am doing. I really appreciate your breaking down of what I am doing 
in my proofs. :) I'll be sure to refer any one I know to ya'lls Web 
site if they have trouble. Talk to you soon.

Thanks,
Victoria