Is it possible to prove that a triangle has 180 degrees, without using parallel line properties?

Question

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.

Answer 1

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 $180^\circ$.)

Answer 2

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 $180^\circ.$