Prove the Parallel Postulate!

All proofs of the Pythagorean Theorem rely essentially on the Parallel Postulate; so if you try to prove what you can in geometry with the Parallel Postulate omitted (this is called 'absolute geometry'), then the Pythagorean Theorem becomes an unprovable theorem within that context. (this is about the only situation where I think the phrase 'uprovable theorem' is the natural choice)

Is it also possible to take the Pythagorean Theorem as an axiom of geometry and prove the Parallel Postulate as a theorem?

• Yes it is! Can you figure out how? – Santana Afton Mar 15 '17 at 10:18
• @JazzyMatrix No, I missed to put the extra question. Can you? – draks ... Mar 15 '17 at 10:19
• here is a good reference. – lulu Mar 15 '17 at 11:07
• @lulu great stuff thanks. As a corollary it would work like (II implies VIII implies IX implies I)... – draks ... Mar 15 '17 at 12:07
• In the modern version of the axioms for a Euclidean plane there are many equivalences to the parallel postulate, including "There exists at least one instance of the theorem of Pythagoras", and "There exists a square" and "There exists a triangle whose interior angles sum to $\pi.$" – DanielWainfleet Mar 15 '17 at 13:57