Plane Geometry - Bruce E. Shapiro

SECTION 31. QUADRILATERALS IN NEUTRAL GEOMETRY 147Theorem 31.4 (Additivity of Angle Sum) Let □ABCD be a convexquadrilateral with diagonal BD. Thenσ(□ABCD) = σ(△ABD) + σ(△BCD)Proof. See figure 31.2. Apply the angle addition postulate (axiom 16.2) toeach of the angles that are split by a diagonal to getσ(□ABCD) = α + β + ɛ + θ= α + γ + δ + ɛ + ζ + η= (α + γ + η) + (δ + ɛ + ζ)+ σ(△ABD) + σ(△BDC)Definition 31.5 The defect of a quadrilateral isδ(□ABCD) = 360 − σ(□ABCD)Theorem 31.6 (Additivity of Defect for Convex Quadrilaterals) If□ABCD is a convex quadrilateral, thenδ(□ABCD) = δ(△ABC) + δ(△ACD)Proof. Apply theorem 31.4.Corollary 31.7 If □ABCD is convex, thenσ(□ABCD) ≤ 360Proof. Apply theorem 31.4 and the Saccheri-Legendre Theorem (theorem30.6).Definition 31.8 □ABCD is called a parallelogram if AB ‖ CD and BC ‖AD.Theorem 31.9 Every parallelogram is convex.Proof. (Exercise)Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.