A Full Axiomatic Development of High School Geometry

This textbook offers a comprehensive axiomatic development of plane Euclidean geometry, covering the standard content of high school geometry. It begins with a set of points, a measure of distance, and ten simple axioms, and concludes with a proof of consistency and a description of their models. It is written for graduate and advanced undergraduate students, pure math majors, faculty, and professional mathematicians, with the aim of anchoring K-12 geometry in solid modern mathematics.

Format: Hardback
Length: 134 pages
Publication date: 14 February 2023
Publisher: Springer International Publishing AG

This comprehensive textbook offers a thorough and systematic axiomatic development of the core aspects of plane Euclidean geometry, commonly taught in high school. It begins with a set of points, a metric defining the distance between pairs of points, and a set of ten fundamental axioms. From these foundational elements, the notions of length, area, angle measure, congruence, and similarity are carefully defined, and their properties are established as theorems. The book concludes with a proof of the consistency of the axioms employed and a comprehensive description of their models. Presented in an inquiry-based format, the book encourages students to actively engage in the learning process, proving the theorems and presenting their proofs to their peers with the instructor serving as a mentor and guide.

Designed for graduate and advanced undergraduate students with an interest in teaching secondary school mathematics, as well as pure math majors seeking to understand the foundations of geometry, this book also serves as a valuable resource for faculty preparing future secondary school teachers and for professional mathematicians seeking a comprehensive reference. The author's primary goal is to anchor K–12 geometry in solid modern mathematics, thereby enhancing the teaching of secondary and tertiary geometry with a deep understanding of the subject.

The textbook is organized into chapters, each dedicated to a specific topic in plane Euclidean geometry. The first chapter introduces the basic elements of points, lines, and planes, while the second chapter explores the properties of angles, such as measures, bisectors, and perpendicular lines. Subsequent chapters delve into the concepts of length, area, and volume, as well as the study of congruence and similarity. The book also includes a chapter on the proof techniques used in geometry, including the use of axiom systems and deductive reasoning.

Throughout the text, numerous examples and exercises are provided to illustrate the concepts and theorems discussed. These examples range from simple to complex, and they are designed to help students grasp the theoretical principles and apply them to real-world situations. Additionally, the book includes a comprehensive glossary of mathematical terms and symbols, as well as a list of references for further reading.

In summary, this textbook provides a comprehensive and accessible introduction to plane Euclidean geometry, suitable for students at various levels of study. With its systematic approach, engaging examples, and comprehensive coverage, it serves as an invaluable resource for anyone interested in deepening their understanding of this fundamental branch of mathematics.

Weight: 395g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031235245
Edition number: 1st ed. 2023