[ahmed2005]

"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]

I think the total degrees of the angles on a triangal is always 180 Degrees, Pi is 3.1416 or something.

[ballengerb1]

"the sum of angles of any triangle is not more than Pi" wher are you getting this theory? Strat is correct on both points.

[Capuchin]

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]

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.

[wolfgangqpublic]

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? :p

[nonAbelian]

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]

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]

The problem is that Stratmando (regardless of whether 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]

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]

Ballanger, I went back, had to give you a balancer.
nonAbelian, Can you tell me what is wrong about my Answers. I don't see the answer in Your Post, except tell them where they can go.
Although it is 4 years old, can you give the answer, we'll wait.
How were your grades in Math?
We can't afford the Book, can you sum it up.

[ebaines]

I think nonAbelian was being a bit harsh. The sum of the interior angles of a triangle in Euclidian geometry is always 180 degrees. The proof of this is really quite simple and is typically covered in first geometry courses (see figure below). However the OP specifically said to not use the 5th axiom of Euclid, which has to do with the fact that non-parallel lines intersect. Strictly speaking the proof in the figure involves relying on alternate interior angles being equal, which in turn is based on the 5th axiom, and hence fails the OP's request.

I think his first "ding" of you was not for not recognizing that in radians is equivaklent to 180 degrees. The second "error" that nonAbelian pointed out is that the sum is 180 degrees only in Euclidian space, meaning "flat" space - the kind of geometery we all learned in elementary school. In curved spaces the sum of angles may be more or less than 180 degrees. Here's an example: suppose you are standing at the Earth's north pole and start walking due south. When you get to the equator you have finished the first leg of your trek - now turn left 90 degrees to head due east, and walk along the equator until you have gone 1/4 of the way around the globe. This completes the second leg. Then turn left 90 degrees again to head due north and walk back to the north pole and your original starting point at the north pole, for the third leg. You will arrive at rigth angles to the first leg of your journey. Thus you have completed the circuit and defined a triangle by walking 3 straight lines with a total of 270 interior degrees. You can get other answers if you change the length of any of the legs.

The surface of a globe is known as "positive curvature" space, and the sum of angles for any three-sided polygon will be greater than 180 degrees on such a surface. There's also a type of space with negative curvature for which the sum of angles is always less than 180 degrees. Negative space resembles a horse sadle. You can learn more about curved spaces here: Euclidean and Non-Euclidean Geometry

But since the OP stated "the sum of the angles is not more than 180 degrees" this rules out positive curvature space, so it's pretty clear to me that he was asking about a proof for Euclidian geometry. Hence nonAbelian's mention of curved spaces is beyond the scope of the original question.

[corrigan]

the question itself is bogus. If you could prove the sum of the angles of a triangle is pi without the fifth axiom, you could prove the fifth axiom using only the first four, and thus negate the need for the fifth axiom. They give field medals for stuff like that.