Plane Geometry - Bruce E. Shapiro

162 SECTION 32. THE EUCLIDEAN PARALLEL POSTULATEBut by equation 32.2∠QT i P < ɛThus we have constructed a point such that the angle α < ɛ, proving thelemma.Figure 32.6: Proof of lemma 32.11.Proof. (The Angle Sum Postulate (axiom 32.10) is equivalent to the EuclideanParallel Postulate (axiom 32.1).)(⇒) [Euclidean Parallel Postulate ⇒ Angle Sum Postulate]Assume the Euclidean Parallel Postulate.Let △ABC be a triangle, and construct point D such that α = β (see figure32.7).Figure 32.7: The Euclidean Parallel Postulate implies the Angle Sum Postulate(axiom 32.10).Then ←→ CD ‖ ←→ AB by the alternate interior angles theorem.Choose E ∈ ←→ CD such that E ∗C ∗D. Since the Euclidean Parallel Postulateimplies the converse to the alternate interior angles theorem, then γ = δ.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012