Table of Contents

1. Equivalence Relations.- 2 Mappings.- 3 The Real Numbers.- 4 Axiom Systems.- One Absolute Geometry.- 5 Models.- 6 Incidence Axiom and Ruler Postulate.- 7 Betweenness.- 8 Segments, Rays, and Convex Sets.- 9 Angles and Triangles.- 10 The Golden Age of Greek Mathematics (Optional).- 11 Euclid’S Elements (Optional).- 12 Pasch’s Postulate and Plane Separation Postulate.- 13 Crossbar and Quadrilaterals.- 14 Measuring Angles and the Protractor Postulate.- 15 Alternative Axiom Systems (Optional).- 16 Mirrors.- 17 Congruence and the Penultimate Postulate.- 18 Perpendiculars and Inequalities.- 19 Reflections.- 20 Circles.- 21 Absolute Geometry and Saccheri Quadrilaterals.- 22 Saccherfs Three Hypotheses.- 23 Euclid’s Parallel Postulate.- 24 Biangles.- 25 Excursions.- Two Non-Euclidean Geometry.- 26 Parallels and the Ultimate Axiom.- 27 Brushes and Cycles.- 28 Rotations, Translations, and Horolations.- 29 The Classification of Isometries.- 30 Symmetry.- 31 HOrocircles.- 32 The Fundamental Formula.- 33 Categoricalness and Area.- 34 Quadrature of the Circle.- Hints and Answers.- Notation Index.