To prove triangle sum theorem, you either have to accept that corresponding angles created by a transversal through two parallel lines are equal or you have to prove that, but when proving that they are equal then you have to accept that the triangle sum is 180 degrees or prove it but that takes me back to where I started from. So my question is, how can we consider these proofs to be proofs if they are codependent. When proving parallel and corresponding theorem, I assumed that two corresponding angles where equal, but the lines weren't parallel, but this requires you to accept that a triangle has 180 degrees angle sum for when two lines aren't parallel and intersected by a transversal then they form a triangle. Are my "proofs" too basic, or am I missing something? For when I try to prove one of them, I have to just accept that the other is a fact, without proving, since you cannot prove any of these without accepting that the other is a fact. Please let me know if you require more information from me. But I just want to know how can I accept these as proofs when I have to accept the other as a fact without proving it.
[Math] Proving triangle sum theorem and parallel lines and corresponding angles
geometry
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Best Answer
Equality of angles follows almost immediately from the original Euclidean version of the parallel postulate: Suppose the corresponding angles are unequal. It follows (given a little calculation) that the sum of the interior angles on one side of the transversal is less than two right angles; hence (by Euclid's version of the postulate) the two lines intersect on that side of the transversal, contradicting the assumption of parallelism. The only 'angle sum' property required is that opposite-side angles always sum to two right angles.