Plane Geometry - Bruce E. Shapiro

SECTION 32. THE EUCLIDEAN PARALLEL POSTULATE 163By the angle addition postulate and the linear sum theorem, α+γ+ɛ = 180.Hence (using α = β and γ = δ we haveσ(△ABC) = δ + β + ɛ = 180Hence the Euclidean Parallel Postulate ⇒ the Angle Sum Postulate.(⇐) [Angle Sum Postulate ⇒ Euclidean Parallel Postulate]Assume the angle sum postulate.Let l be a line and P ∉ l a point. Drop a perpendicular from P to l andcall the foot of the perpendicular Q (see figure 32.8).Let m be the line through P that is perpendicular to ←→ P Q. By the alternateinterior angle theorem m ‖ l.Suppose that there is a second line n through P such that n ≠ m and n ‖ l(RAA).Choose S on n such that S ∈ H Q,m .Choose R on m such that R ∈ H ←→ S, P Q.Choose T on l such that T ∈ H ←→ S, P Qand ɛ = ∠QT P < ∠SP R = γ. Sucha point exists by the lemma because n ‖ l.Point T is in the interior of ∠QP S (Otherwise, if S were in the interior of∠QP T , then n would intersect QT by the Crossbar theorem, which is notpossible since they are parallel.)Henceby the protractor postulate.α < δSince S is in the interior of ∠β, we also haveγ + δ = β = 90Since l ‖ m, the alternate interior angles are equivalent, i.e.,∠T P S = ɛ =⇒ α + ɛ < δ + γ = 90also by the protractor postulate, and consequentlyσ(△P QT ) = α + ɛ + 90< δ + γ + 90= 180Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.