At the college level, Geometry is not taught for its applications. It is taught either for its beauty, or as a way to teach proof techniques and systems of axioms. In this book, it is the latter concept that prevails. Students who already had a course on proofs will not need these chapters. The book ends with an extensive Appendix, in which systems of axioms dominate one more time.

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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. Students are prepared to teach high school geometry by making connections that enable a deeper understanding of the subject.

Comprehensive coverage of most of Euclid's Elements includes almost all of the material in the first six books of that work. Careful statements of the axioms encourage students to understand how the theorems of geometry are built on the axioms.

Coverage of transformations and the transformational approach to the foundations contains a complete classification of rigid motions of the plane and an explanation of how the Reflection Postulate can substitute for the Side-Angle-Side Postulate. This helps students understand the transformational perspective and how it can be incorporated into the axioms.

A complete construction of the traditional models for Hyperbolic Geometry provides a model that helps students understand the relationships spelled out in the axioms. A study of geometry in the real world examines some non-traditional models and curved spaces, helping students arrive at answers to questions that arise naturally about the relationship between non-Euclidean geometry and the geometry of the real world.

Careful attention to the historical development of geometry discusses certain cultural and philosophical issues, and demonstrates the changes that the foundations of geometry have gone through over time.

An introduction to proof acts as a bridge between lower-level courses in which technique is emphasized to upper-level courses in which proof and the understanding of concepts are emphasized. This enables students to experience the axiomatic method and study careful proofs that characterize more advanced mathematics.

Emphasis on how to write proofs provides many model proofs as well as commentary on key proofs. Provides students with an excellent introductory chapter on how to write proofs, and provides more advanced techniques as they progress through the course.

Includes many proofs that are exercises, with hints given so that students know roughly how the proof should be structured. Enables students to concentrate on how to organize and communicate the proof. Earlier presentation of neutral geometry has been achieved by shifting some topics to appendices and covering others more efficiently. For example, material on set theory and the real numbers was moved to an appendix because much of it is review for most students.

The review of proof writing has been incorporated into the chapter on Axiomatic Systems. Description of different axiom systems for elementary geometry has been moved to its own appendix previously found at the beginning of the chapter on The Axioms of Plane Geometry. Instructors can now cover this material at any time they choose during the course.

A new Appendix B Systems of Axioms for Geometry explores in depth the range of options available in the choice of axioms and explains the rationale for the choices made in this book. Foundations of Geometry Zoom. Product information Instructor Resources Description 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.

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. New to this Edition Earlier presentation of neutral geometry has been achieved by shifting some topics to appendices and covering others more efficiently.

Table of Contents 1. Prologue: Euclid's Elements 1. Axiomatic Systems and Incidence Geometry 2. Axioms for Plane Geometry 3.

Neutral Geometry 4. Euclidean Geometry 5. Hyperbolic Geometry 6. Area 7. Circles 8. Constructions 9. Transformations Models Polygonal Models and the Geometry of Space Euclid's Book I A. Systems of Axioms for Geometry B. The Postulates Used in this Book C. Set Notation and the Real Numbers D. Your Basket.

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## Foundations of Geometry

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.

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