A Question Relating Sum of Angles and Breaking of the Parallel Postulate

[walkeraj]

Question: What is the relationship between the sum of the angles of a non-euclidean triangle being greater or less than 180 degrees and the definite breaking of the parallel postulate? Is the proof of this trivial?

Edit: Additionally, can we say that if the angles of a triangle sum to greater or less than 180 degrees it can be shown that the parallel postulate has been broken?

Edit 2: Can the above edit be shown in a method Euclid himself might have employed?

 

[jbriggs444]

Question: What is the relationship between the sum of the angles of a non-euclidean triangle being greater or less than 180 degrees and the definite breaking of the parallel postulate? Is the proof of this trivial?

You want a direct, constructive proof? Because a proof by contradiction is immediate -- given the parallel postulate, you get standard Euclidean geometry where the angles of a triangle are provably 180 degrees. If a triangle does not satisfy that and if all of the other postulates hold good then the Parallel Postulate is falsified.

 

[WWGD]

I think the sum of angles equaling $\pi$ is equivalent to being in flat, i.e., non-curved space. Though that doesn't follow from Euclid, I don't think.

 

[Hill]

The proof of this equivalency, and several others, is provided here.

 

[pbuk]

Question: What is the relationship between the sum of the angles of a non-euclidean triangle being greater or less than 180 degrees and the definite breaking of the parallel postulate? Is the proof of this trivial?

Edit: Additionally, can we say that if the angles of a triangle sum to greater or less than 180 degrees it can be shown that the parallel postulate has been broken?

Edit 2: Can the above edit be shown in a method Euclid himself might have employed?

There seems to be some confusion here.

Euclid proved, using the so-called "parallel postulate" as an axiom, that the angles of a triangle sum to 180°. The proof is not trivial, but it is quite simple and can be found easily on line.

Therefore he also proved (by [Edit]inversion contraposition) that if in some geometry the interior angles of a triangle do not sum to 180° then the parallel postulate is false in that geometry. That is why we call such geometries non-Euclidean.

 

[Hill]

There seems to be some confusion here.

Euclid proved, using the so-called "parallel postulate" as an axiom, that the angles of a triangle sum to 180°. The proof is not trivial, but it is quite simple and can be found easily on line.

Therefore he also proved (by inversion) that if in some geometry the interior angles of a triangle do not sum to 180° then the parallel postulate is false in that geometry. That is why we call such geometries non-Euclidean.

This proves only half of the equivalency: if the fifth postulate is true, then sum of angles of any triangle is $\pi$.
There is the other half: if sum of angles of any triangle is $\pi$, is the statement of the fifth postulate true?

 

[pbuk]

There is the other half: if sum of angles of any triangle is $\pi$, is the statement of the fifth postulate true?

Oops, that is indeed the inverse of the sum of angles theorem but I meant to refer to the contrapositive, not the inverse. Corrected, thanks.