I'm having trouble proving this equivalence. All I can get is that if the sum of the interior angles of any triangle is $180^\circ$, then Euclid's fifth postulate is valid.
When I try to generalize the sum of the angles of a triangle, I get stuck in the proof. I've looked in some books about this demonstration, but none provide a similar explanation beyond stating the sum as $180^\circ$, without offering a generalized approach.