# Proof of a theory about angles of triangle

"the sum of angles of any triangle is not more than Pi" I want to know the proof of this statement without using the fifth axiom of Euclid.Thank you

 Stratmando Posts: 10,437, Reputation: 2520 Uber Member #2 Apr 13, 2008, 12:35 PM
I think the total degrees of the angles on a triangal is always 180 Degrees, Pi is 3.1416 or something.
nonAbelian (Nov 17, 2011 05:34 AM):   Source:
 ballengerb1 Posts: 25,644, Reputation: 11296 Home Repair & Remodeling Expert #3 Apr 13, 2008, 01:10 PM
"the sum of angles of any triangle is not more than Pi" wher are you getting this theory? Strat is correct on both points.
nonAbelian (Nov 17, 2011 05:34 AM):   Source:
 Capuchin Posts: 5,319, Reputation: 3601 Uber Member #4 Apr 13, 2008, 01:22 PM
Ehm, I would assume that the statement would be using radians, where 360 degrees is 2pi radians. So, the angles inside a triangle sum to exactly pi radians.

I'm not sure how to prove this though, it seems rather obvious.
 galactus Posts: 2,272, Reputation: 1436 Ultra Member #5 Apr 13, 2008, 02:49 PM
Actually, you can do this with a piece of paper.

Cut a triangle out of a piece of paper, then tear off a piece that includes each vertex.

Reassemble the pieces so that the vertices coincide. Then, take a ruler and place it along side. It should be a straight line coinciding with Pi or 180 degrees.
Attached Images

 wolfgangqpublic Posts: 192, Reputation: 140 - #6 Dec 17, 2008, 03:02 PM
Lol, the fifth axiom of Euclid.

The statement should be:
"the sum of angles of any triangle is exactly Pi"

Basic elementary school proof+Basic degrees to radians conversion=fifth axiom of Euclid?
 nonAbelian Posts: 3, Reputation: -3 Junior Member #7 Nov 17, 2011, 05:33 AM
The sum of the angles in a triangle is not always pi. It can be more or less depending on the curvature of the surface that the triangle lies on. This is a corollary of the famous Gauss-Bonnet theorem. According to G-B the sum of the angles is equal to pi (or "180 degrees") only when the curvature of the surface is zero, more than pi when the curvature is positive, and less than pi when the surface has negative curvature. Also, pi is most definitely not 3.1416. It has a non-terminating decimal expansion since pi is irrational (in fact, it is transcendental).
ballengerb1 (Nov 17, 2011 06:55 AM):   Source:
massplumber2008 (Nov 25, 2011 07:24 AM):   Source:
 ballengerb1 Posts: 25,644, Reputation: 11296 Home Repair & Remodeling Expert #8 Nov 17, 2011, 07:15 AM
So yet another one post wonder pops up and gives bad marks on a 4 year old post. Interesting how you gave me a bad mark for asking a question. BTW Stratmando's two comments are correct. You may not think he answered the question but he was correct. Where were you 4 years ago??
 nonAbelian Posts: 3, Reputation: -3 Junior Member #9 Nov 22, 2011, 05:37 PM
The problem is that Stratmando (regardless of whether or not he answered the question) is not correct on either account. Don't just take my word for it; read a university level textbook on math. In particular, pick up a book on differential geometry for a solid exposition on this subject. The book by Wolfgang Kuhnel (Differential Geometry: Curves-Surfaces-Manifolds) should suffice.
 nonAbelian Posts: 3, Reputation: -3 Junior Member #10 Nov 22, 2011, 05:43 PM
To answer your last question though 4 years ago I was studying for my degree in Applied Math. I found this page during a search which quite honestly was not related to this question, but it seemed like the people on here were not qualified to answer the question.
Stratmando (Sep 4, 2012 12:21 PM): NA   Source:

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