What Are Proofs?

Date: 08/12/97 at 02:39:14
From: Femail
Subject: Geometry proof

I am a home school student through American schools, and am stuck in
geometry.  I do not understand proofs.  Can you help me out? 

Thank you.

S. Schmit

Date: 08/18/97 at 09:09:44
From: Doctor Rob
Subject: Re: Geometry proof

The study of geometry is the first place I encountered an axiom 
system.

You start with certain "undefined objects," in this case "point,"
"line," "plane," "length," "area," "between," etc. Then you are given
certain statements about them which you are to accept as true. These
are called Postulates or Axioms. They appear in the very first part of
your book on Plane Geometry. Examples might be:

   Any two distinct points lie on one and only one line.

   Any two lines contain at most one common point.

   Given a line and a point not on it, there is one and only one line
   not intersecting the given line and containing the given point.

The idea is now to develop as many true statements (called theorems)
as you can about points, lines, etc., from these few postulates by
using logical reasoning. A proof is a detailed description of the
logical reasoning used to deduce a theorem from either the postulates
or other previously proven theorems.

Along the way, more objects will be defined in terms of the original
undefined objects, such as an angle, a line segment, a triangle, a
circle, and so on. Theorems about these new objects are also deduced
in the same way.

The first theorems you prove from the postulates are very simple
statements. Using them, you prove more and more complicated theorems,
including the one which says that the sum of the measures of the
interior angles in a triangle is 180 degrees, and the Pythagorean
Theorem (the sum of the squares of the lengths of the legs of a right
triangle equals the square of the length of the hypotenuse).

One confusing thing about geometry proofs is that there is no single
correct answer. Any logical sequence of true statements which ends
with the desired theorem is a valid proof. Always there are many
correct proofs of any theorem. Some are shorter and simpler than
others, and those are usually preferred by teachers, students, and
mathematicians. Some are clever, some are tedious, and some are even
considered beautiful or elegant. (I'll bet you didn't think that
esthetics could enter into mathematics, but it definitely does!)

Often proofs are constructed by working backwards. Starting with the
desired conclusion T, you could say, "If I could prove statement A,
then using previously proved theorem (or postulate) B, I could
conclude that T is true." This reduces your proof to proving
statement A, then saying at the end of that proof, "Using Theorem B,
T is true." Often there are many possibilities for A (and B). The
trick is to pick one you can prove! Often several different plausible 
choices for A (and B) are tried to find one which works (for you).

Some useful hints: Review carefully the hypothesis and conclusion of
the theorem you want to prove. Keep in mind the postulates and
previously proved theorems which might apply to the situation at hand.
Consider working backwards, as described above. Draw a figure (or
several) to illustrate the situation. Consider constructing helping
lines, points, circles, etc., which might be useful.

I hope that this is of some use to you.  If you have specific 
questions, write back and we'll try to help you find solutions.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 08/19/97 at 00:13:44
From: femail
Subject: Re: Geometry proof

Thank you! I will give it a shot and get back to you if I have further
problems.