Is there really no proof to corresponding angles being equal?

Question

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?

Answer 1

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 $180^{\circ}$.