Euclid's Fifth Postulate

Date: 6/24/96 at 20:30:17
From: Joshua Minsley
Subject: Euclid's Fifth Postulate

Dear Dr. Math,

I am interested in finding some theorems, axioms, postulates, 
or whatever that are similar to Euclid's Fifth Postulate 
(the Parallel postulate).

Any help you can offer, or directions you can point me in, would 
be wunderbar!

   Sincerely,
               Joshua M. Minsley

Date: 6/25/96 at 15:4:50
From: Doctor Peter
Subject: Re: Euclid's Fifth Postulate

I am assuming that you are looking for equivalent statements to the
Parallel Postulate, that is statements that can be proven using the 
the five postulates, but cannot be proven without the fifth.

An example would be that if two lines are parallel, the alternate
interior angles formed by the transversal are congruent.  

Another statement that relies on the parallel postulate is that the 
sum of the angle measures of a triangle is 180 degrees.  

This discussion will quickly head towards the differences between 
Euclidean (with the parallel postulate) and Non-Euclidean (no parallel 
postulate) geometries.  However, if you would like to read more into 
this, I would refer you to Chapter 5 of Greenberg's Euclidean and Non-
Euclidean Geometries.  You will find a ton of equivalencies to Euclid 
V in there.  

Please write back if you have more questions.

Doctor Peter - The Math Forum
Check out our website! http://mathforum.org/dr.math/   

Date: 6/26/96 at 0:31:43
From: Joshua Michael Minsley
Subject: Thanks...

Ahhh... the bitter irony!  The book you recommended is my textbook for 
Axiomatic Geometry here at Michigan State University.

I was actually searching the WWW for some equivalent theorems, when I
happened upon your Web page...  Thank you for the help.
                Josh Minsley