Plane Geometry - Bruce E. Shapiro

SECTION 32. THE EUCLIDEAN PARALLEL POSTULATE 165Figure 32.9: The Euclidean Parallel Postulate imples Wallis’ Postulate.Figure 32.10:Wallis’ Postulate implies the Euclidean Parallel Postulate.By the angle sum postulateHenceσ(ABC) = σ(DEF ) = 180ν = 180 − (γ + δ)= 180 − (α + β)= µand △ABC ∼ △DEF . Hence the Euclidean Parallel Postulate ⇒ Wallis’Postulate.(⇐) (Wallis’ Postulate ⇒ the Euclidean Parallel Postulate). Assume Wallis’Postulate is true.Let l be a line and P ∉ l a point. Drop a perpendicular from P to its footQ on l. Construct m through P perpendicular to ←→ P Q (see figure 32.10).Let n be the line through P that is parallel to l.Suppose n ≠ m (RAA).Chose S ∈ m such that S ∈ H Q,m . Let R be the foot of the perpendicularto ←→ P Q dropped from S.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.