Foundations of geometry /
Venema, Gerard.
Boston, Mass. ; London : Pearson,
c2012.
Synopsis:  Foundations of Geometry, Second Edition is written to help enrich the education of all mathematics majors and facilitate a smooth transition into more advanced mathematics courses. The text also implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers-and encourages students to make connections between their college courses and classes they will later teach. This text's coverage begins with Euclid's Elements, lays out a system of axioms for geometry, and then moves on to neutral geometry, Euclidian and hyperbolic geometries from an axiomatic point of view, and then non-Euclidean geometry. Good proof-writing skills are emphasized, along with a historical development of geometry. The Second Edition streamlines and reorganizes material in order to reach coverage of neutral geometry as early as possible, adds more exercises throughout, and facilitates use of the open-source software Geogebra. This text is ideal for an undergraduate course in axiomatic geometry for future high school geometry teachers, or for any student who has not yet encountered upper-level math, such as real analysis or abstract algebra. It assumes calculus and linear algebra as prerequisites.
Includes bibliographical references and index.
Preface -- 1:  Prologue:  Euclid's Elements -- 1-1:  Geometry before Euclid -- 1-2:  Logical structure of Euclid's Elements -- 1-2:  Logical structure of Euclid's Elements -- 1-3:  Historical significance of Euclid's Elements -- 1-4:  Look at Book I of the Elements -- 1-5:  Critique of Euclid's Elements -- 1-6:  Final observations about the Elements -- 2:  Axiomatic Systems And Incidence Geometry -- 2-1:  Structure of an axiomatic system -- 2-2:  Example:  incidence geometry -- 2-3:  Parallel postulates in incidence geometry -- 2-4:  Axiomatic systems and the real world -- 2-5:  Theorems, proofs, and logic -- 2-6:  Some theorems from incidence geometry -- 3:  Axioms For Plane Geometry -- 3-1:  Undefined terms and two fundamental axioms -- 3-2:  Distance and the ruler postulate -- 3-3:  Plane separation -- 3-4:  Angle measure and the protractor postulate -- 3-5:  Crossbar theorem and the Linear Pair theorem -- 3-6:  Side-angle-side postulate -- 3-7:  Parallel postulates and models -- 4:  Neutral Geometry -- 4-1:  Exterior Angle theorem and existence of perpendiculars -- 4-2:  Triangle congruence conditions -- 4-3:  Three inequalities for triangles -- 4-4:  Alternate interior angles theorem -- 4-5:  Saccheri-Legendre theorem -- 4-6:  Quadrilaterals -- 4-7:  Statements equivalent to the Euclidean parallel postulate -- 4-8:  Rectangles and defect -- 4-9:  Universal hyperbolic theorem -- 5:  Euclidean Geometry -- 5-1:  Basic theorems of Euclidean geometry -- 5-2:  Parallel projection theorem -- 5-3:  Similar triangles -- 5-4:  Pythagorean theorem -- 5-5:  Trigonometry -- 5-6:  Exploring the Euclidean geometry of the triangle -- 6:  Hyperbolic Geometry -- 6-1:  Basic theorems of hyperbolic geometry -- 6-2:  Common perpendiculars -- 6-3:  Angle of parallelism -- 6-4:  Limiting parallel rays -- 6-5:  Asymptotic triangles -- 6-6:  Classification of parallels -- 6-7:  Properties of the critical function -- 6-8:  Defect of a triangle -- 6-9:  Is the real world hyperbolic? -- 7:  Area -- 7-1:  Neutral area postulate -- 7-2:  Area in Euclidean geometry -- 7-3:  Dissection theory -- 7-4:  Proof of the dissection theorem in Euclidean geometry -- 7-5:  Associated Saccheri quadrilateral -- 7-6:  Area and defect in hyperbolic geometry -- 8:  Circles -- 8-1:  Circles and lines in neutral geometry -- 8-2:  Circles and triangles in neutral geometry -- 8-3:  Circles in Euclidean geometry -- 8-4:  Circular continuity -- 8-5:  Circumference and area of Euclidean circles -- 8-6:  Exploring Euclidean circles --
9:  Constructions -- 9-1:  Compass and straightedge constructions -- 9-2:  Neutral constructions -- 9-3:  Euclidean constructions -- 9-4:  Construction of regular polygons -- 9-5:  Area constructions -- 9-6:  Three impossible constructions -- 10:  Transformations -- 10-1:  Properties of isometries -- 10-2:  Rotations, translations, and glide reflections -- 10-3:  Classification of Euclidean motions -- 10-4:  Classification of hyperbolic motions -- 10-5:  Transformational approach to the foundations -- 10-6:  Similarity transformations in Euclidean geometry -- 10-7:  Euclidean inversions in circles -- 11:  Models -- 11-1:  Cartesian model for Euclidean geometry -- 11-2:  Poincare disk model for hyperbolic geometry -- 11-3:  Other models for hyperbolic geometry -- 11-4:  Models for elliptic geometry -- 12:  Polygonal Models And The Geometry Of Space -- 12-1:  Curved surfaces -- 12-2:  Approximate models for the hyperbolic plane -- 12-3:  Geometric surfaces -- 12-4:  Geometry of the universe -- 12-5:  Conclusion -- 12-6:  Further study -- 12-7:  Templates -- Appendices: -- A:  Euclid's Book I -- A-1:  Definitions -- A-2:  Postulates -- A-3:  Common notions -- A-4:  Propositions -- B:  Systems of axioms for geometry -- B-1:  Hilbert's axioms -- B-2:  Birkoff's axioms -- B-3:  MacLane's axioms -- B-4:  SMSG axioms -- B-5:  UCSMP axioms -- C:  Postulates used in this book -- C-1:  Criteria used in selecting the postulates -- C-2:  Statements of the postulates -- C-3:  Logical relationships -- D:  Van Hiele model -- E:  Set notation and the real numbers -- E-1:  Some elementary set theory -- E-2:  Axioms for the real numbers -- E-3:  Properties of the real numbers -- E-4:  One-to-one and onto functions -- E-5:  Continuous functions -- F:  Hints for selected exercises -- Bibliography -- Index.
Synopsis:  Foundations of Geometry, Second Edition is written to help enrich the education of all mathematics majors and facilitate a smooth transition into more advanced mathematics courses. The text also implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers-and encourages students to make connections between their college courses and classes they will later teach. This text's coverage begins with Euclid's Elements, lays out a system of axioms for geometry, and then moves on to neutral geometry, Euclidian and hyperbolic geometries from an axiomatic point of view, and then non-Euclidean geometry. Good proof-writing skills are emphasized, along with a historical development of geometry. The Second Edition streamlines and reorganizes material in order to reach coverage of neutral geometry as early as possible, adds more exercises throughout, and facilitates use of the open-source software Geogebra. This text is ideal for an undergraduate course in axiomatic geometry for future high school geometry teachers, or for any student who has not yet encountered upper-level math, such as real analysis or abstract algebra. It assumes calculus and linear algebra as prerequisites.
Geometry