Plane Geometry - Bruce E. Shapiro

150 SECTION 31. QUADRILATERALS IN NEUTRAL GEOMETRYTheorem 31.13 If □ABCD is a non-convex quadrilateral then □ACBDis a quadrilateral.Proof. Since □ABCD is a quadrilateral no three of the points A, B, C, Dare collinear.Since □ABCD is a quadrilateral BC ∩ AD = ∅.Since □ABCD is not convex then AC and BD are disjoint (the diagonalsdo not intersect).Thus segments AC, CB, BD, and DA share at most their endpoints.Hence □ACBD is a quadrilateral.Definition 31.14 □ABCD is a Saccheri Quadrilateral if ∠ABC =∠BAD = 90 and AD = BC. Segment AB is called the base and segmentDC is called the summit.Figure 31.5: A Saccheri Quadrilateral. It is not possible to prove that thisfigure is a rectangle using only the axioms of neutral geometry - one mustaccept Euclid’s fifth postulate to do so. Thus we sometimes draw the topand side edges as curves to remind ourselves of this.Theorem 31.15 The diagonals of a Saccheri Quadrilateral are congruent.Proof. Consider triangles △ABD and △ABC. Since BC = AD, AB =AB, and ∠A = 90 = ∠B, they are congruent. Hence BD ∼ = AC.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012