# Is there really no proof to corresponding angles being equal?

I've read in this question that the corresponding angles being equal theorem is just a postulate. However I find this unsatisfying, and I believe there should be a proof for it. However the only way that I can think of proving it is by a proof by contradiction involving two lines that are not parallel, and a transcendental such that the corresponding angles $x$ and $y$ are equal. However this would require knowledge of the fact that the sum of angles in a triangle is equal to $180^\circ$. So consequently, is there a way to prove that the sum of angles in a triangle are equal to $180^\circ$ without using corresponding, alternating, or supplementary angles postulates?

• See the answer to this post for several axiomatizations of Euclid's geometry. – Mauro ALLEGRANZA Jan 6 '16 at 12:21

No, because without the parallel postulate (which these others derive from) you could have some non-Euclidean geometry where the sum of angles in a triangle is not $180^{\circ}$.

• so such an unintuitive theorem is just made a postulate? – Airdish Jan 6 '16 at 12:12
• It depends who you ask. It can be proven from the parallel postulate (see the comments in the second answer of your link), but the parallel postulate can't be proven. There are various ways to take a set of axioms to begin with. (also I wouldn't personally call this unintuitive) – Morgan Rodgers Jan 6 '16 at 12:22