Triangle Proofs in General

Date: 11/19/2001 at 20:49:48
From: Ashley
Subject: Triangle proofs

Hello, 

I'm having a difficult time with proofs in general. I am able to do 
the really simple proofs but as they get harder and longer I get 
really lost, especially with the method cpctc after you prove 
triangles congruent. I am able to start the proof but am never able to 
complete it. 

I will show you an example of one I am having difficulty with:  

                                                     C
                                                    /  \
                                                   /    \
given: AB is a straight line, ca~=cb,<1~=<2       /1 /\2 \
prove: <3~=<4                                    /  /  \  \
                                                /  /3  4\  \
                                               A-D-----E---B-

I know that the first step has to be the given. But then I don't know 
where to go from there. Should I prove the first triangle congruent 
(tri. acd~=tri. bcd)? And if I do, how do I get the third side 
congruent. Because we already have a side and an angle, therefore is 
it SAS, because we need the other side? 

If you can help and understand what I am asking, that would be greatly 
appreciated. Thanks so much.

Ashley

Date: 11/19/2001 at 23:40:55
From: Doctor Peterson
Subject: Re: Triangle proofs

Hi, Ashley.

We have a FAQ on proofs that contains the sort of thing I am about to 
tell you, and more:

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

A lot of people write to tell us they get stuck as soon as they have 
written down the "givens." I think that's something like "writer's 
block"; let's call it "prover's block." What do you do when you sit 
down in front of a blank sheet of paper, or a pair of empty columns?

Let's imagine another situation in which you have no idea what to do: 
you've just parachuted from a disabled plane and find yourself in 
unknown country, not knowing just where you are or how to get back to 
civilization. What would you do?

I would probably first climb a tree or otherwise get the best view I 
can of the territory. In your case, you might start by drawing the 
picture and marking the known facts, as you have; then look it over 
and familiarize yourself with it. Think not only about what you are 
told, but about what you see, since too often textbooks don't do a 
good job of stating everything in the "givens," and force you to make 
some assumptions from the picture, such as that D and E are on segment 
AB, and the points are in the order ADEB. Look for triangles that 
MIGHT be congruent; you can't just assume that, and the picture might 
be deliberately drawn to hide such things (if you didn't draw your own 
picture, which is always a good idea!), but it helps to have an idea 
what you might be able to prove true. In this example, ACD and BCE are 
a good choice.

Here's my personal copy of the picture, on which if I had room I would 
mark the known pairs of congruent sides or angles:

            C
            +
          // \\
        /1/   \2\
      /  /3   4\  \
    +---+-------+---+
   A    D       E    B

I see from the fact that AC=BC that triangle ACB is isosceles, and 
that carries with it some extra information, such as that the base 
angles are equal. That's the sort of thing you want to run through 
your mind as you look around the problem.

Next I would decide where I want to go, and choose a possible route. I 
might know that there is a town near the lake, and therefore decide to 
find a stream and follow it downstream. I wouldn't expect to know 
exactly where to go without a map, but I would want to have some 
general ideas of the direction I'm heading. In your proof, you want 
angles 3 and 4 to be equal; what "lake" is that fact on the "shore" 
of? It sounds like the same fact I mentioned before: the base angles 
of an isosceles triangle. So if I can prove that DCE is an isosceles 
triangle, I will be able to get to the goal.

What I've really done so far is to think from both ends of the 
problem: where can I go from where I am (the givens), and how might I 
get to the goal?

Now it's time to start trudging along. I'll never get anywhere unless 
I start moving; even if I don't know just where to go, I may run into 
something (like a stream running downhill) that I can follow. You 
might take a pair of triangles that look congruent and make a list of 
their "corresponding parts" to see if you have enough to prove 
congruence. (Don't assume it has to be SAS; there's always ASA.) You 
can use both given facts and implications that you noticed as you 
looked around.

One way to help yourself get started is to avoid writing anything down 
in the official columns yet; you're just scouting around, and don't 
have to get anything in the right order yet. Just jot down any ideas 
you have. If you like writing (or if you don't), you may have some 
favorite methods of organizing thoughts for an essay; use any method 
you want to keep track of your ideas.

Eventually, something will click, and you'll realize that you have 
what you need to get at least a good part of the way toward the goal. 
Write that down in a half-orderly way so you won't forget it, and then 
look around again.

I've avoided being too precise here to encourage you to go at it 
yourself; but there should be enough here to help you move forward. If 
you need more help with this problem, please write back and show me 
how far you are able to get, even if it's just disorganized ideas. 

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

Date: 11/20/2001 at 20:41:23
From: Ashley
Subject: Re: Triangle proofs

Thank you so much for the advice. I went back and tried this problem 
after I wrote you and I did something similar to what you were saying. 
I usually like to look at the picture I drew and think of the things 
in it like if there are vertical angles, supplementary angles, 
complementary angles etc., and see how I can go from there. I got a 
lot further than I had when I wrote you and I think that I did pretty 
well.