Euclid's Fifth PostulateDate: 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 |
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