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Parallel Lines: Two Column Proof

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Date: 09/09/98 at 23:06:05
From: Turtle
Subject: Geometry

Hello,

I was wondering if there is any way that you could break down the steps
in doing a two-column proof? One that we had to do for homework is:

Given: Angle 1 congruent angle 2, angle 3 congruent angle 4
Prove: n parallel p

l         m
/        3/
------/---------/------n
/         /
1 /        2/
----/---------/------p    angle 5 is under angle 2
/4        /
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Date: 09/10/98 at 17:04:39
From: Doctor Peterson
Subject: Re: Geometry

Hi, Turtle.

Two-column proofs are a little foreign to most of us - even to
mathematicians, who don't usually use such a rigid way of writing a
proof once they have learned what it means to prove something. The idea
is to force you to think very clearly and express yourself very
precisely. Unfortunately, no one really thinks that way, so if you're
just shown a two-column proof without an explanation of how someone
produced it, it seems like either magic ("how did he do that?") or a
waste of time ("why did he bother to do all that?").

I suggest that you first try to prove your goal without thinking about
the details of the two columns. People too often get bogged down in
the details ("what is the exact reason for this step?" "is this
legal?") when the important thing about a proof is to learn the logic
behind it.

A proof is sort of a bridge from the "mainland" of known truth to an
"island" you want to get to. In your case, you have been given a
platform you are supposed to start from (the "givens"), and you have
some set of definitions, postulates, and already-proven theorems that
you can use. Think of those as materials you can use to build the
bridge:

Given: angle 1 congruent angle 2
angle 3 congruent angle 4
Prove: n parallel p

l         m
/        3/
------/---------/------n
/         /
1 /        2/
----/---------/------p
/4        /

Now you need to take a quick helicopter ride over the territory between
the starting and ending points and see what you can recognize as useful
stepping stones. It will take some practice to get used to what sorts
of things you should look for. Just make a list of facts you either can
deduce from the givens or can use to get to the goal. This is like
building a bridge by starting at both ends and working toward the
middle:

Givens:
Angle 1 = angle 2
Angle 3 = angle 4

Deductions (from the givens):
Angle 1 = angle 4 (vertical angles)
Angle 1 = angle 2 implies that l and m are parallel

Possibilities (for proving the goal):
n is parallel to p if angles 2 and 3 are equal

I have 2 = 1 and 1 = 4 and 4 = 3, so I can prove that 2 = 3.
(Notice I don't need all of my deductions)

Now you have the idea of a proof, and you can start working out the
details. We can locate the stepping stones our bridge will use:

Given:
Angle 1 = angle 2
Angle 3 = angle 4
Steps:
Angle 1 = angle 4 (vertical angles)
2 = 3 (because 2 = 1 = 4 = 3)
n is parallel to p (because 2 = 3)

It's starting to look like a proof. But now we need to organize it by
stating clearly just what the reason is for each step - laying boards
between the stepping stones - and filling in some steps that help
clarify what we are doing:

Given:
Angle 1 congruent angle 2
Angle 3 congruent angle 4

Statement:                     | Reason:
-------------------------------+--------------------------
1: Angle 1 congruent angle 4  | vertical angles are equal
2: Angle 1 congruent angle 2  | given
3: Angle 2 congruent angle 4  | both equal to angle 1
4: Angle 3 congruent angle 4  | given
5: Angle 2 congruent angle 3  | both equal to angle 4
6: Line n is parallel to p    | transversal makes equal angles

The exact wording will depend on what your text gives as the names of
the theorems you use, how your teacher asks you to lay out the proof,
and so on. That isn't the important thing, and you should not worry
about the "rivets" in your proof (as long as they pass the teacher's
inspection). Think about it; you've just built a bridge to new
territory. When you get further, you'll have much more significant
proofs to write. None of them will be a Golden Gate, but some will be
pretty impressive. Right now, you're just hopping a puddle, and it
might not look like much, but it's good practice for building the big
bridges.

Here's a nice brief explanation of this process in our Dr. Math
archives, if you'd like another perspective:

http://mathforum.org/dr.math/problems/2_col_proofs.html

http://mathforum.org/dr.math/problems/crystal9.12.98.html

I hope this helps you learn to write proofs.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry

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