Geometry

1 Preliminary Notions.- 1.1 Axioms and Models.- 1.2 Sets and Equivalence Relations.- 1.3 Functions.- 2 Incidence and Metric Geometry.- 2.1 Definition and Models of Incidence Geometry.- 2.2 Metric Geometry.- 2.3 Special Coordinate Systems.- 3 Betweenness and Elementary Figures.- 3.1 An Alternative Description of the Euclidean Plane.- 3.2 Betweenness.- 3.3 Line Segments and Rays.- 3.4 Angles and Triangles.- 4 Plane Separation.- 4.1 The Plane Separation Axiom.- 4.2 PSA for the Euclidean and Hyperbolic Planes.- 4.3 Pasch Geometries.- 4.4 Interiors and the Crossbar Theorem.- 4.5 Convex Quadrilaterals.- 5 Angle Measure.- 5.1 The Measure of an Angle.- 5.2 The Moulton Plane.- 5.3 Perpendicularity and Angle Congruence.- 5.4 Euclidean and Hyperbolic Angle Measure (optional).- 6 Neutral Geometry.- 6.1 The Side-Angle-Side Axiom.- 6.2 Basic Triangle Congruence Theorems.- 6.3 The Exterior Angle Theorem and Its Consequences.- 6.4 Right Triangles.- 6.5 Circles and Their Tangent Lines.- 6.6 The Two Circle Theorem (optional).- 6.7 The Synthetic Approach (optional).- 7 The Theory of Parallels.- 7.1 The Existence of Parallel Lines.- 7.2 Saccheri Quadrilaterals.- 7.3 The Critical Function.- 8 Hyperbolic Geometry.- 8.1 Asymptotic Rays and Triangles.- 8.2 Angle Sum and the Defect of a Triangle.- 8.3 The Distance Between Parallel Lines.- 9 Euclidean Geometry.- 9.1 Equivalent Forms of EPP.- 9.2 Similarity Theory.- 9.3 Some Classical Theorems of Euclidean Geometry.- 10 Area.- 10.1 The Area Function.- 10.2 The Existence of Euclidean Area.- 10.3 The Existence of Hyperbolic Area.- 10.4 Bolyai's Theorem.- 11 The Theory of Isometries.- 11.1 Collineations and Isometries.- 11.2 The Klein and Poincar Disk Models (optional).- 11.3 Reflections and the Mirror Axiom.- 11.4 Pencils and Cycles.- 11.5 Double Reflections and Their Invariant Sets.- 11.6 The Classification of Isometries.- 11.7 The Isometry Group.- 11.8 The SAS Axiom in ?.- 11.9 The Isometry Groups of ? and ?.