Plane Geometry - Bruce E. Shapiro

Section 34The Parallel ProjectionTheoremLemma 34.1 Let l, m, and n be distinct parallel lines; let t and t ′ betransversals that cut these three lines at points A, B, C, and A ′ , B ′ , C ′ withA ∗ B ∗ C. ThenAB = BC ⇒ A ′ B ′ = B ′ C ′Figure 34.1: Proof of theorem 34.1. This is a special case of the ParallelProjection Theorem (theorem 34.2) when AB = BC.Proof. By the Euclidean Parallel Postulate, we can construct lines t ′′ andt ′′′ parallel to t through A ′ and B ′ , as shown in figure 34.1.173