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 Poincaré 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 ?.

This book is intended as a first rigorous course in geometry. As the title indicates, we have adopted Birkhoff's metric approach (i.e., through use of real numbers) rather than Hilbert's synthetic approach to the subject. Throughout the text we illustrate the various axioms, definitions, and theorems with models ranging from the familiar Cartesian plane to the Poincare upper half plane, the Taxicab plane, and the Moulton plane. We hope that through an intimate acquaintance with examples (and a model is just an example), the reader will obtain a real feeling and intuition for non­ Euclidean (and in particular, hyperbolic) geometry. From a pedagogical viewpoint this approach has the advantage of reducing the reader's tendency to reason from a picture. In addition, our students have found the strange new world of the non-Euclidean geometries both interesting and exciting. Our basic approach is to introduce and develop the various axioms slowly, and then, in a departure from other texts, illustrate major definitions and axioms with two or three models. This has the twin advantages of showing the richness of the concept being discussed and of enabling the reader to picture the idea more clearly. Furthermore, encountering models which do not satisfy the axiom being introduced or the hypothesis of the theorem being proved often sheds more light on the relevant concept than a myriad of cases which do.