# 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?

• In hyperbolic non- euclidean geometry where lines are geodesic and corresponding angles equal they are parallel. Any two conditions imply the third. – Narasimham Jul 22 '15 at 5:30
• Narasimham, thank you very much for your commnt!! Your comment is complicated for me... But, in Euclidean geometry, is a postulate needed to prove two lines are parallel? Why does one book start with a postulate, and another book never uses a postulate? – chris vin Jul 22 '15 at 15:09
• @Narasimham If your comment were an answer I would downvote it. It doesn't answer the question and introduces complications that don't help illuminate the problem. – Simon S Jul 22 '15 at 15:14
• Let me put it in perspective. Yes, I too don't justify it. That we are so close to fifth postulate that was invoked for non-euclidean geometry is a matter not entirely irrelevant here. However, I shall delete the comment soon.if someone had told me the water is deep here at an early stage, it would be remembered for long as a watershed situation. – Narasimham Jul 22 '15 at 20:58

• @chris, yes, you can take (2) as a postulate instead of Euclid's fifth postulate, but that's a matter of choice. Most textbooks take instead as a postulate the so called "uniqueness of parallel": given a line $r$ and an external point $P$, there is only one line in the plane passing through $P$ and parallel to $r$. From that you can prove (2). – Aretino Jul 23 '15 at 7:10