Plane Geometry - Bruce E. Shapiro

SECTION 31. QUADRILATERALS IN NEUTRAL GEOMETRY 153Proof. Since □ABCD is convex (theorem 31.19), σ(□ABCD) ≤ 360 (Corollary31.7). By definition of a Saccheri Quadrilateral, ∠A = ∠B = 90. Hence∠C + ∠D ≤ 180By theorem 31.16 the summit angles are congruent, hence ∠C = ∠D.Hence ∠C ≤ 90 and ∠D ≤ 90.Definition 31.21 A Lambert quadrilateral is a quadrilateral in whichthree of the interior angles are right angles.Figure 31.8: A Lambert Quadrilateral. Like the Saccheri Quadrilateral, tis not possible to prove that this figure is a rectangle using only the axiomsof neutral geometry - one must accept Euclid’s fifth postulate to do so.Corollary 31.22 Let □ABCD be a Lambert quadrilateral. Then it is aparallelogram.Proof. By construction of the Lambert quadrilateral, DC ⊥ BC and AB ⊥BC hence either DC ‖ AB or DC = AB. Since the two sides are distinct,that means they are parallel.Similarly, by construction, AD ⊥ AB and AB ⊥ BC. Since AD and BCare distinct, they must therefore be parallel.Corollary 31.23 Let □ABCD be a Lambert quadrilateral.convex.Then it isProof. It is a parallelogram and all parallelograms are convex.Corollary 31.24 Let □ABCD be a Lambert quadrilateral with right anglesat vertices A, B, and C. Then ∠D is either a right angle or it isacute.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.