Ideal for those who may have little previous experience with abstraction and proof, this book provides a rigorous and unified -- yet straightforward and accessible -- introduction to the foundations of Euclidean, hyperbolic, and spherical geometry. It helps readers develop analytic and reasoning skills and gain an awareness of the depth, power and subtlety of the axiomatic method in general, and of Euclidean and non-Euclidean plane geometry in particular. Covers: The Question of Parallels; Five Examples; Some Logic; ...
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Ideal for those who may have little previous experience with abstraction and proof, this book provides a rigorous and unified -- yet straightforward and accessible -- introduction to the foundations of Euclidean, hyperbolic, and spherical geometry. It helps readers develop analytic and reasoning skills and gain an awareness of the depth, power and subtlety of the axiomatic method in general, and of Euclidean and non-Euclidean plane geometry in particular. Covers: The Question of Parallels; Five Examples; Some Logic; Practice Proofs; Set Terminology and Sets of Real Numbers; An Axiom System for Plane Geometry: First Steps; Betweeness, Segments and Rays; Three Axioms for the Line; The Real Ray Axiom and Its Consequences; Antipodes and Opposite Rays; Separation; Pencils and Angles; The Crossbar Theorem; Side-Angle-Side; Perpendiculars; The Exterior Angle Inequality and Triangle Inequality; Further Inequalities Concerning Triangles; Parallels and the Diameter of the Plane; Angle Sums of Triangles; Parallels and Angle Sums; Concurrence Theorems; Circles; and Similarity.
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