AMSCO'S Geometry. New York - Rye High School

CHAPTERQUADRILATERALSEuclid’s fifth postulate was often considered to bea “flaw” in his development of geometry. GirolamoSaccheri (1667–1733) was convinced that by the applicationof rigorous logical reasoning, this postulatecould be proved. He proceeded to develop a geometrybased on an isosceles quadrilateral with two baseangles that are right angles.This isosceles quadrilateralhad been proposed by the Persian mathematicianNasir al-Din al-Tusi (1201–1274).Using this quadrilateral,Saccheri attempted to prove Euclid’s fifth postulate byreasoning to a contradiction. After his death, his workwas published under the title Euclid Freed of Every Flaw.Saccheri did not, as he set out to do, prove the parallelpostulate, but his work laid the foundations for newgeometries. János Bolyai (1802–1860) and NicolaiLobachevsky (1793–1856) developed a geometry thatallowed two lines parallel to the given line, through apoint not on a given line. Georg Riemann (1826–1866)developed a geometry in which there is no line parallelto a given line through a point not on the given line.10CHAPTERTABLE OF CONTENTS10-1 The General Quadrilateral10-2 The Parallelogram10-3 Proving That a QuadrilateralIs a Parallelogram10-4 The Rectangle10-5 The Rhombus10-6 The Square10-7 The Trapezoid10-8 Areas of PolygonsChapter SummaryVocabularyReview ExercisesCumulative Review379