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Two-column proofs

Date: 18 Dec 1994 
From: Marvin W. Osterbur
Subject: geometry proofs

I am writing on behalf of my daughter Mel who is a sophomore in
high school. She is having a real problem with proofs. In 
particular two column proofs. Can you explain the steps to prove
geometric figures? I don't know how to word this any better so I
hope you can figure out what information we are trying to get.
We wrote to someone else and basically got a definition of what
a proof is. Thanks for any help that you can give.

Date: Mon, 19 Dec 1994 
From: Stephen Weimar
Subject: Re: geometry proofs

Most of the doctors have gone home for the break but I'll make a 
quick attempt to answer your question, although I suspect that we 
need to hear/read more about the way your daughter is working on 
proofs in order to be really helpful.

A proof is meant to take the reader from a hypothetical to a conclusion,
showing why we should have no doubt of the truth.

  A\ w/E
   / z\
 D/    \B

Hypothetical: If segments AB and DE intersect, 
Conclusion: then the opposite angles (x&y, w&z) formed by the intersection
are equal.

After stating the hypothetical situation, begin building your proof with
what you know.  Usually there are some conditions that are given or there
are some basic postulates you can rely on. 

The proof is built like a kind of scaffolding; once you state a premise
and show why that's true, then you can confidently make another assertion
which is supported by the previous premise.

With a two-column proof you write what you know on the left side and on
the right you say how you know that.

The difficulty with two-column proofs often seems to revolve around how
people organize their thinking and the two-column proof does not fit with
the way many of us think.  It only serves to confuse our thinking.  If
this is the case then one solution is to first sketch out the proof in our
own words and then put it into the two-column format. 

Another common difficulty is that the steps may seem so obvious it's hard
to know what is needed for a proof, or why a proof is needed.

Take our vertical angle example. This is something with which we are so
familiar that it may be hard for us to recover the proof that we've
already done in our heads long ago.  When you can't figure out what needs
to be said it often helps to start speaking out loud and ask yourself "how
do you know?" after each statement you make.

"It's obvious that the angles are equal."

"Yes. How do you know?"

"Because x and z make up a straight line and so do z and y."

"Right, so how do you know x and y are equal?"

"If x and z is the same as y and z then take z away and you're left with x
equals y."

"Great!  That's a proof, now you just have to write it out in two columns."

AB intersects DE forming            Hypothesis
vertical angles x and y, z and w

x+z=180 degrees                     definition of supplementary
y+z=180 degrees                     ""
x+z=y+z                             substitution
x=y                                 subtraction

Hope this is useful.  Please write back with follow-up questions.

-- steve ("chief of staff")
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry

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