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Please make the proof as simple as possible; I'm still a beginner. Or rather, it's fine if you make it complex, but try to explain it using layman's language.

Ethan Chan
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    No, because it is not true in geometries that don't have the parallel line property (Euclid's fifth postulate.) – Thomas Andrews Jun 04 '18 at 13:28
  • @ThomasAndrews Even going beyond Euclid (who, frankly, were in the right track, but couldn't hold up to the late 19th century rise in demand for mathematical rigor), the parallel postulate is an important part of geometric systems such as the one by Hilbert for defining what we intuitively know as Euclidean geometry. Without it, we can prove the existence part of the parallel postulate, but not the uniqueness. We can prove that triangles have an angle sum of at most $180^\circ$, but not exactly $180^\circ$. – Arthur Jun 04 '18 at 13:32
  • I dpn't see what the point is of your comment. What point do you think i am making? @Arthur – Thomas Andrews Jun 04 '18 at 13:34
  • @ThomasAndrews Putting too much faith in Euclid as the basis for geometry (that was just the bracketed part, though). Apart from that, it's not a response to you, but rather expanding on your comment. – Arthur Jun 04 '18 at 13:35
  • @Arthur That's not what I said at all. My argument is entirely a mathematical one. There are things which satisfy all of the postulates other than the parallel postulate, and, in some of those, there are triangles which do not add up to 180 degrees. So you can't prove it with only the other axioms. What "faith" has to do with it is unclear to me. – Thomas Andrews Jun 04 '18 at 13:38
  • @ThomasAndrews I think you put too much weight on a relatively small and somewhat unimportant part of my original comment. I noted that you mentioned Euclid, so I thought it worthwhile to point out that technically, Euclid is outdated and there are more modern axiomatic approaches. But even though we've managed to do "better" than Euclid, the parallel postulate is still indispensable. That was my main point. – Arthur Jun 04 '18 at 13:44
  • You recently asked essentially the same question here, where I suggested an answer.https://math.stackexchange.com/questions/2807341/are-the-proofs-for-the-properties-of-parallel-lines-and-that-a-triangle-has-180 . That is really bad manners on this site, because it wastes the time of people who try to help. I will vote to close this as a duplicate. – Ethan Bolker Jun 04 '18 at 13:54

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The essential way to prove "Theorem A can't be proved in axiom set X" is to find a "thing" (called a model) which satisfies the axiom set X in which A is not true.

There are models, like the Poincaré disk model, in which have "points," "lines," "angles," etc., which we can prove satisfy all the properties of Euclidean geometry other than the parallel postulate.

In the Poincaré model, the angles of a triangle always add up to less than $180^\circ.$

Thomas Andrews
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If it was possible, then the theorem would be true on the surface of a sphere. But it is not. It's easy to define a triangle there such the sum of all its internal angles is $270^\circ$, for instance. (Acutally, the sum will always greater than $180^\circ$.)

José Carlos Santos
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  • The sphere does not satisfy the rest of Euclid's postulates, though, since there are multiple lines (great circles) for anti-podal points. – Thomas Andrews Jun 04 '18 at 13:33
  • @ThomasAndrews I know that, but it is still possible (and it is likely, in my opinion) that this example is enough for the OP. However, if the OP says the same thing as you, I will delete it. – José Carlos Santos Jun 04 '18 at 13:34
  • I think the Poincaré disc, while possibly not as well known, would be a more correct counterexample, as we there actually do satisfy all the other axioms of Euclidean geometry (even more modern systems than Euclid's own). – Arthur Jun 04 '18 at 13:40