Plane Geometry - Bruce E. Shapiro

SECTION 32. THE EUCLIDEAN PARALLEL POSTULATE 157Suppose they intersect on the opposite side. Then △P QC has exteriorangles α and β that form linear pairs with interior angles 180 − α and180 − β. PicK either linear pair, say the one formed with α. Then bythe exterior angle theorem α > 180 − β, which contradicts the fact thatα + β < 180. Hence the lines must intersect on the same side as α and β.This is Euclid’s 5th postulate.(⇐) (E5P ⇒ EPP). Assume that Euclid’s 5th postulate is true.Let l be a line and P be a point such that P ∉ l.Drop a perpendicular line from P to l, and call the foot of the line Q.Construct m through P such that m ⊥ ←→ P Q. By the alternate interior anglestheorem (theorem 29.4) l ‖ m.Assume n ≠ m is a second line through P such that n ‖ l.Then ←→ P Q is a transversal to n and l. Since n ≠ m, the interior anglesγ ≠ 90 and δ ≠ 90. Since they form a linear pair, γ + δ = 180.Hence one of γ, δ is less than 90 and the other is greater than 90.By Euclid’s fifth postulate, lines n and l meet on whichever side of ←→ P Q thesmaller of angles γ and δ lies (e.g., on the same side as δ in figure 32.2).Thus n ̸‖ l. Hence there is only one line through P that is parallel to l.Hence the Euclidean Parallel Postulate follows from Euclid’s Fifth Postulate.Figure 32.2: Euclid’s Fifth Postulate implies the Euclidean Parallel Postulate.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.