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Getting Started with Two-Column Proofs


Date: 03/19/2002 at 22:05:35
From: Kayla
Subject: Getting started with Two-column Proofs

Dear Dr. Math,

I am really stuck with this Two-Column Proof thing. I am home schooled 
and I basically teach myself. I understand it a little bit but 
sometimes I can't get started on my own. I am really a numbers person 
and can't solve anything or prove anything without numbers. Here is 
an example of a question.

Write a two-column proof of Theorem 8-1, which says: If the altitude 
is drawn from the vertex of the right triangle to its hypotenuse, then 
the two triangles formed are similar to the given triangle and to each 
other.

How am I supposed to prove that? How do I get started?

Kayla


Date: 03/19/2002 at 22:53:13
From: Doctor Peterson
Subject: Re: Getting started with Two-column Proofs

Hi, Kayla.

I'll be happy to help; we get a lot of questions from others in your 
position, because proofs are something new to most students. You'll 
want to read through our FAQ on the subject, if you haven't already:

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

You call yourself a "numbers person"; do you feel comfortable with 
writing or reasoning as well? That is really closer to the idea of 
proofs than all the work with numbers that you've done in earlier 
math, and thinking of a proof as a kind of persuasive essay is a 
better image of what you have to do. Your job in a proof is to 
convince me that something is true, as long as I accept the postulates 
you're basing it on.

Let's see how I would approach this particular proof:

First, we have to draw a picture, so we can see what it's about:

         B
         +
        /| \
       / |   \
      /  |     \
     +---+-------+
    A    D        C

Here AC is the hypotenuse and B is the right angle, so we drew 
altitude BD, which is perpendicular at D. I've named all the points to 
help me talk about them; and on paper I would mark little squares to 
show that angles ABC, ADB, and CDB are right angles.

That's what we're given, together with some immediate implications 
(such as which angles are right). Now, what is it we have to prove? 
The "two triangles formed" must be ABD and CBD; they are supposed to 
be similar to ABC. But before we can say this symbolically, we have to 
decide what orientation to use; that is, in what order should we name 
the vertices of each triangle, so we are matching up the right sides 
when we call them similar? Well, if they are similar, then 
corresponding angles will be congruent, and triangles ABC and ABD 
share one angle at A, so A must match in both. And both have a right 
angle, at B and D respectively. So one of our conclusions must be "ABC 
is similar to ADB." In the same way, the other conclusion must be "ABC 
is similar to BDC."

Now, in just thinking enough to decide what it is we have to prove, 
I've actually discovered enough to go ahead and write the proof. See 
if you can do that, then write back and show me what you come up with. 
We can go through it together, and get you on the right track with 
proofs.

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

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