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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?

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  • $\begingroup$ In hyperbolic non- euclidean geometry where lines are geodesic and corresponding angles equal they are parallel. Any two conditions imply the third. $\endgroup$ – Narasimham Jul 22 '15 at 5:30
  • $\begingroup$ 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? $\endgroup$ – chris vin Jul 22 '15 at 15:09
  • $\begingroup$ @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. $\endgroup$ – Simon S Jul 22 '15 at 15:14
  • $\begingroup$ 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. $\endgroup$ – Narasimham Jul 22 '15 at 20:58
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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).

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  • $\begingroup$ needing an additional postulate to prove theorem (2) above is a great info!! Thank you very much. Then, is it safe to say that (2) is like a postulate, and that (1) is a theorem since it can be proved by contradiction? $\endgroup$ – chris vin Jul 23 '15 at 1:04
  • $\begingroup$ Thank you very much for making my questions more clear!! $\endgroup$ – chris vin Jul 23 '15 at 1:08
  • $\begingroup$ @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). $\endgroup$ – Aretino Jul 23 '15 at 7:10
  • $\begingroup$ your answer is very clear!! Now I understand it. Thank you very much for the additional answer!! $\endgroup$ – chris vin Jul 23 '15 at 18:39

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