Equivalents to the Euclidean Parallel PostulateIn this section we work within neutral geometry to prove that a numberof different statements are equivalent to the Euclidean ParallelPostulate (EPP). This has historical importance. We noted before thatEuclid’s Fifth Postulate was quite different in tone and complexityfrom his first four postulates, and that Euclid himself put off using thefifth postulate for as long as possible in his development of geometry.The history of mathematics is rife with efforts to either prove the fifthpostulate from the first four, or to find a simpler postulate from whichthe fifth postulate could be proved. These efforts gave rise to anumber of candidates that, in the end, proved to be no simpler than thefifth postulate and in fact turned out to be logically equivalent, at leastin the context of neutral geometry. Euclid’s fifth postulate indeedcaptured something fundamental about geometry. We begin with the converseof the Alternate Interior Angles Theorem,as follows:If two parallel lines nand m are cut by a transversal l, thenalternate interior angles are congruent. We noted earlier that this statement was equivalent to the EuclideanParallel Postulate. We now prove that. Theorem:The converse of the Alternate Interior Angles Theorem isequivalent to the Euclidean Parallel Postulate.~We first assume the converse of the Alternate Interior AnglesTheorem and prove that through a given line l and a point P not on theline, there is exactly one line through P parallel to l. We have alreadyproved that one such line must exist: Drop a perpendicular from P to l;call the foot of that perpendicular Q. Let . Now, throughpoint P use the protractor postulate to construct a line m perpendicularto t. The Alternate Interior Angles Theorem guarantees that since thealternate interior angles are all right angles, land mare parallel. So itall comes down to proving that there is only one such line.
To this end, suppose there were another line mNthrough P parallel to l. By the converse to the Alternate Interior Angles Theorem, alternateinterior angles formed by l, mN, and tmust be equal, and since tzl wemust have mNztas well. But then mand mNwould both beperpendiculars to tthough point P, which would violate the uniquenessof rays forming a right angle with tthrough a given point (ProtractorPostulate). Thus, m= mN. To prove the converse, assume that the EPP holds and that land mareparallel lines cut by a transversal tat points Q and P, respectively. Weprove that alternate interior angles are congruent. Suppose forcontradiction that alternate interior angles p1 (at point P) and p2 (atpoint Q) are not congruent, and WLOG that :(p1) > :(p2). Using theprotractor postulate, create ray on the other side of tfrom p2 suchthat :(pRPQ) = :(p2). Then from the above theorem, since pRPQand p