Plane Geometry - Bruce E. Shapiro

160 SECTION 32. THE EUCLIDEAN PARALLEL POSTULATETheorem 32.6 The following are equivalent to the Euclidean Parallel Postulate.1. Hilbert’s Parallel Postulate. For every line l and for every externalpoint P there exists at most one line m such that P ∈ m andm ‖ l.2. Proclus’ Axiom. If l ‖ m and t ≠ l is a line such that t insersectsl then t also intersects m.3. If l ‖ m and t is a transversal such that t ⊥ l then t ⊥ m.4. if l, m, n and k are lines such that k ‖ l, m ⊥ k, and n ⊥ l, theneither m = n or m ‖ n.5. Transitivity of Parallels. If l ‖ m and m ‖ n then eitiher l = n orl ‖ n.Proof. (Exercise.)Transitivity of parallelism has the following two corollaries that we will uselater.Corollary 32.7 The diagonals of a parallelogram are congruent.Corollary 32.8 The diagonals of a parallelogram bisect one another.Proof. (Exercise.)Theorem 32.9 There exists a rectangle ⇐⇒ the EPP is true.Proof. (Given in a later section.)Figure 32.5: Illustration of lemma 32.11. There is a point T ∈ −→ QR suchthat α < ɛ.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012