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### Thinking About Proofs

```
Date: 09/24/97 at 15:29:27
From: Erika
Subject: PROOFS

How do you know what statement to write next when you're doing a
proof?  And what are the reasons that you use?

ex. given:  m<4 + m<6 = 180
prove:  m<5 = m<6                           K
/ \
/   \
/     \
/       \
____4_/5______6_\
J          L
```

```
Date: 09/24/97 at 18:03:11
From: Doctor Pete
Subject: Re: PROOFS

Hi,

Probably the most difficult part of proving something is where to
start. It takes a lot of practice, and trial and error. Since you are
more interested in the thought process than the solution, let me tell
you what went through my mind as I solved your example.

"Well, I don't need to know anything about the lengths of the
triangle, because all I've been given are angles. I remember that
the sum of the angles of a triangle is always 180 degrees, but
that doesn't help because I don't know what angle 5 is nor do I
know what angle JKL is. Well, I know that angle 4 + angle 6 = 180.
Ah! But I also know that angle 4 + 5 = 180, because together they
form the straight line which contains JL. So if angle 4 + angle 6
= 180 = angle 4 + angle 5, then angle 4 + angle 6 = angle 4 +
angle 5, and once we subtract angle 4 from both sides of this
equation, we get our result, angle 6 = angle 5."

Notice that I tried to rule out some possible ways of proving the
theorem, based on what I was given (and importantly, what I was not
given). I then relied on previous theorems or things I already knew
that were relevant. I applied what I was given in the problem to this
previous knowledge, and I was able to determine that some of my ideas
didn't work. Luckily, I recognized that I could apply one of my ideas
to the solution, and at that point I was able to see that this was the
right direction to take. The rest was a matter of finishing the
details.

While this is not always how proofs are made, it gives you an idea of
the process, which is more or less "educated trial-and-error."  You
guess, you eliminate possibilities, narrow it down, and push the line
of reasoning further. Some proofs, like this one, only required one
"breakthrough" idea, namely, the fact that supplementary angles add to
180 degrees. However, some proofs require applying many ideas, all in
the right order. That's why they're difficult, because at each point
in the proof there are many directions to take. Knowing which is the
right one is something that takes practice and patience.

-Doctor Pete,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
Middle School Geometry
Middle School Two-Dimensional Geometry

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