It is well known that the Pythagorean theorem is equivalent (in the context of neutral or absolute geometry) to Euclid's Fifth postulate. It is also true that the converse of the Pythagorean theorem is equivalent to Euclid's fifth, but I have never seen a proof of this. Does anybody know how to prove that the converse of Pythagoras implies Euclid's Fifth or any known equivalent such as Playfair's postulate or the fact that the sum of the angles of a triangle is 180 degrees?
Euclid's proof, in Elements I, 48, of the converse of the Pythagorean Theorem [I, 47] is a reductio proof resting on I, 47. And since I, 47 (thru I, 41 & 29) rests on Postulate 5, the converse I, 48 also rests on Postulate 5.