Plane Geometry - Bruce E. Shapiro

134 SECTION 29. TRANSVERSALSFigure 29.2: Angles α and β are corresponding angles.Theorem 29.4 (Alternate Interior Angles Theorem) Suppose that land m are cut by transversal t in such a way that a pair of alternate interiorangles are congruent. Then l ‖ m.Proof. Suppose β = γ in figure 29.1. Suppose there is a point P that lies onboth l and m. Because t is a transversal, it intersects l and m at differentpoints; hence P ∉ t. Therefore by the plane separation postulate (axiom15.2) it must be on one side or the other of t, as illustrated in figure 29.3. IfFigure 29.3: Proof of alternate interior angle theorem (theorem 29.4). Left:case when P ∈ H C,t ; Right: case when P ∈ H A,t . Both cases violate theexterior angle theorem (theorem 24.4)P ∈ H C,t then γ is an exterior angle of triangle △P BE while β is a remoteinterior angle (corresponding to γ) of the same triangle.By the exterior angle theorem (theorem 24.4), γ > β; Since β = γ this is acontradiction, hence P ∉ H C,t .If P ∈ H A,t , then β is an exterior angle of triangle △P BE, while γ is aremote interior angle (corresponding to β) of the same triangle. This alsoviolates the exterior angle theorem.Hence there is no such point P , and that means the lines l and m neverintersect. So they are parallel.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012