Equivalents to the Euclidean Parallel Postulate.pdf -...

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Equivalents to the Euclidean Parallel Postulate In this section we work within neutral geometry to prove that a number of different statements are equivalent to the Euclidean Parallel Postulate (EPP). This has historical importance. We noted before that Euclid’s Fifth Postulate was quite different in tone and complexity from his first four postulates, and that Euclid himself put off using the fifth postulate for as long as possible in his development of geometry. The history of mathematics is rife with efforts to either prove the fifth postulate from the first four, or to find a simpler postulate from which the fifth postulate could be proved. These efforts gave rise to a number of candidates that, in the end, proved to be no simpler than the fifth postulate and in fact turned out to be logically equivalent, at least in the context of neutral geometry. Euclid’s fifth postulate indeed captured something fundamental about geometry. We begin with the converse of the Alternate Interior Angles Theorem, as follows: If two parallel lines n and m are cut by a transversal l , then alternate interior angles are congruent. We noted earlier that this statement was equivalent to the Euclidean Parallel Postulate. We now prove that. Theorem: The converse of the Alternate Interior Angles Theorem is equivalent to the Euclidean Parallel Postulate. ~ We first assume the converse of the Alternate Interior Angles Theorem and prove that through a given line l and a point P not on the line, there is exactly one line through P parallel to l. We have already proved that one such line must exist: Drop a perpendicular from P to l ; call the foot of that perpendicular Q. Let . Now, through point P use the protractor postulate to construct a line m perpendicular to t . The Alternate Interior Angles Theorem guarantees that since the alternate interior angles are all right angles, l and m are parallel. So it all comes down to proving that there is only one such line.
To this end, suppose there were another line m N through P parallel to l . By the converse to the Alternate Interior Angles Theorem, alternate interior angles formed by l , m N , and t must be equal, and since t z l we must have m N z t as well. But then m and m N would both be perpendiculars to t though point P, which would violate the uniqueness of rays forming a right angle with t through a given point (Protractor Postulate). Thus, m = m N . To prove the converse, assume that the EPP holds and that l and m are parallel lines cut by a transversal t at points Q and P, respectively. We prove that alternate interior angles are congruent. Suppose for contradiction that alternate interior angles p 1 (at point P) and p 2 (at point Q) are not congruent, and WLOG that : ( p 1) > : ( p 2). Using the protractor postulate, create ray on the other side of t from p 2 such that : ( p RPQ) = : ( p 2). Then from the above theorem, since p RPQ and p

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