I Some Historical Background. - 1. The Parallel-Postulate Problem. - 2. The Lost Centuries. - 3. Saccheri and the Near Miss. - 4. The Correct PerspectiveGauss Bolyai Lobatchevsky. - 5. Not-Euclidean Geometry Absolute Geometry. - II Absolute Plane Geometry. - 1. Linear Sets and Linear Order. - 2. Half-Planes Angles and Angle Measure. - 3. Triangle Relations Congruence Foot in a Set. - 4. Non-intersecting Lines. - 5. Dedekind Cut Continuity A Basic Circle Property. - 6. Motions and Symmetries. - III Hyperbolic Plane Geometry. - 1. Hyperbolic Parallels. - 2. Biangles Hyperparallels. - 3. Saccheri and Lambert Quadrilaterals Polygon Angle Sums. - 4. Angle of Parallelism Function Triangle Defect Distance Variations. - 5. The 3-Point Property Cycles. - 6. Hyperbolic Compass and Straight Edge Constructions. - 7. Existence Problems; the Method of Associated Right Triangles. - IV A Euclidean Model of the Hyperbolic Plane. - 1. An Overview of the Model. - 2. Circular Inversions in E2. - 3. Angle and Cross Ratio Invariance Under Inversion. - 4. Linear Order and Motions in the Model. - 5. Half-Planes Angles and Angle Measure in the Model. - 6. Triangle Congruence in the Model the Consistency of Hyperbolic Geometry. - Appendix Distance Geometrics. - Topic I. Metric Space and Metric Geometry. - Topic II. A Spherical Metric. - Topic III. Elliptic Geometry. - Topic IV. Barbilian Geometries the Cross Ratio Metric. Language: English