Plane Geometry - Bruce E. Shapiro

158 SECTION 32. THE EUCLIDEAN PARALLEL POSTULATEAxiom 32.4 (Converse of Alternate Interior Angles Theorem.) Iftwo parallel lines are cut by a transversal, then both pairs of alternateinterior angles are congruent.Theorem 32.5 The Converse of the Alternate Interior Angles Theorem isEquivalent to the Euclidean Parallel Postulate.Proof. (⇒) [ Assume that the converse of the Alternate Interior AnglesTheorem is true (Equivalent Axiom 32.4) and show that the EuclideanParallel Postulate (axiom 32.1) follows. ]Let l be a line and let P be a point such that P ∉ l.By theorem 21.5 we can drop a perpendicular line t from P to l and thenconstruct a second line m through P such that t ⊥ m.By the alternate interior angles theorem m ‖ l (because a pair of interiorangles are congruent; it just happens that by construction, all of the interiorangles are all right angles, figure 32.3).Suppose there is another line n ≠ m such that P ∈ n and n ‖ l.Then t is a transversal to l and n. Then by axiom 32.4 the alternate interiorangles δ = γ. Since γ = 90 then δ = 90, and consequently n ⊥ t.By the uniqueness part of the protractor postulate, there is only one linethrough P that is perpendicular to t. Hence n = m.Thus there is only one line parallel to l through P . Thus axiom 32.1 follows.Figure 32.3: The converse of the alternate interior angles theorem (converseto theorem 29.4) is equivalent to the Euclidean Parallel Postulate.(⇐) [ Assume the Euclidean Parallel Postulate (axiom 32.1) and show that« CC BY-NC-ND 3.0. Revised: 18 Nov 2012