Plane Geometry - Bruce E. Shapiro

Section 32The Euclidean ParallelPostulateIn this section we consider some statements that are equivalent to Euclid’s5th postulate.When we say that two statements are equivalent in this sense we meanthat if we add either statement to the axioms of Neutral Geometry, we canprove the other statement. It does not mean that the two statements areprecisely logically equivalent.Axiom 32.1 .Euclidean Parallel Postulate (EPP). For every line l andfor every point P that does not lie on l there is exactly one line m suchthat p ∈ m and m ‖ l.Axiom 32.2 Euclid’s Fifth Postulate (E5P). Let l and m be two linescut by a transversal in such a way that the sum of the measures of the twointerior angles on one side of t is less than 180. Then l and m intersect onthat side of t.Theorem 32.3 Euclid’s 5th postulate ⇐⇒ the Euclidean Parallel Postulate.Proof. (⇒) (EPP ⇒ E5P). Assume that the EPP is true.Let l, m, t, α, β be as indicated in figure 32.1, i.e., construct the lines l,m, and n as shown, with then α + β < 180. We need to show that lines land m intersect on the same side of t as α and β.There is a line n through B such that γ + δ = 180 (by the protractor155