Plane Geometry - Bruce E. Shapiro

154 SECTION 31. QUADRILATERALS IN NEUTRAL GEOMETRYProof. Since □ABCD is a convex quadrilateral then σ(□ABCD) ≤ 360.Since the sum of the first three angles is 270, the remaining one must be≤ 90.Corollary 31.25 Let □ABCD be a Lambert quadrilateral with right anglesat vertices A, B, and C. Then BC ≤ AD.Figure 31.9: In a Lambert quadrilateral, BC ≤ ADProof. Suppose BC > AD (RAA).Then there exists a point E with B ∗ E ∗ C such that BE = AD (rulerpostulate).□ABED is a Saccheri quadrilateral (def. of Saccheri quadrilateral).Hence ∠BED ≤ 90 (theorem 31.20).Angle ∠BED is an exterior angle for △ECD.Angle ∠C = 90, and it is a remote angle of ∠BED, By the exterior angletheorem (theorem 24.4), ∠C < ∠BED (strict inequality). This leads to theresult 90 < 90; therefore we must reject the RAA hypothesis and concludethat BC ≤ AD.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012