Plane Geometry - Bruce E. Shapiro

Section 33RectanglesDefinition 33.1 A rectangle is a quadrilateral each of whose angles areright angles.Definition 33.2 A square is a rectangle each of whose sides has equallength.Theorem 33.3 The following statements are equivalent to the Euclideanparallel postulate:1. There exists a triangle whose defect is 0.2. There exists a right triangle with defect 0.3. (Clairut’s Axiom.) There exists a rectangle.4. There exists arbitrarily large rectangles.5. The defect of every right triangle is 0.6. The defect of every triangle is 0.Lemma 33.4 Let △ABC be a triangle. Then at least two of its interiorangles are acute.Proof. Suppose that two of the interior angles are not acute. Call theseangles α and β, and call the third interior angle δ.Then α ≥ 90 and β ≥ 90. Henceσ(△ABC) = α + β + δ≥ 180 + δBut by the Saccheri-Legendre Theorem (theorem 30.6) σ(△ABC) ≤ 180.Hence δ = 0, which is impossible. Therefore at most one of the angles canbe non-accute, making the remaining two acute.167