Does proving that two lines are parallel require a postulate? Harold Jocobs' Geometry book(2nd Ed) has a Theorem that states "If two lines form equal corresponding angles with a transversal, then the lines are parallel," and gives a indirect proof. He assumes that the lines are not parallel and shows this assumption leads to a contradiction(since if the lines intersect, the angles are not congruent). Another textbook(McDougal Littell's Geometry) have Corresponding Angles Postulate that says "If two parallel lines are cut by a transversal then the pairs of corresponding angles are congurent." The two statements are converse, but Jacobs' book doesn't use a postulate to prove other parallel lines theorems. Sould the Corresponding Angles Postulate be a theorem, and not a postulate? If it can be proved by indirect proof, shouldn't it be just a theorem like the one in Jacobs' book? I understand the indirect proof of Jacobs' Theorem, but why do other books use a postulate?

You are considering two **different** theorems:

(1) If two lines form equal corresponding angles with a transversal, then the lines are parallel.

(2) If two lines are parallel, then they form equal corresponding angles with a transversal.

In euclidean geometry, you need an additional postulate to prove theorem (2) (the famous "Euclid's fifth postulate"), while that is not needed to prove theorem (1).