Foundations of Geometry

ISBN-10: 0136020585
ISBN-13: 9780136020585
Edition: 2nd 2012 (Revised)
Authors: Gerard Venema
List price: $117.80 Buy it from $69.42 Rent it from $30.68
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Description: Foundations of Geometry, Second Editionimplements 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  More...

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Book details

List price: $117.80
Edition: 2nd
Copyright year: 2012
Publisher: Addison Wesley
Publication date: 7/6/2011
Binding: Paperback
Pages: 408
Size: 7.75" wide x 9.50" long x 0.50" tall
Weight: 1.738
Language: English

Foundations of Geometry, Second Editionimplements 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 Editionstreamlines 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.

Prologue: Euclid's Elements
Geometry before Euclid
The logical structure of Euclid's Elements
The historical significance of Euclid's Elements
A look at Book I of the Elements
A critique of Euclid's Elements
Final observations about the Elements
Axiomatic Systems and Incidence Geometry
The structure of an axiomatic system
An example: Incidence geometry
The parallel postulates in incidence geometry
Axiomatic systems and the real world
Theorems, proofs, and logic
Some theorems from incidence geometry
Axioms for Plane Geometry
The undefined terms and two fundamental axioms
Distance and the Ruler Postulate
Plane separation
Angle measure and the Protractor Postulate
The Crossbar Theorem and the Linear Pair Theorem
The Side-Angle-Side Postulate
The parallel postulates and models
Neutral Geometry
The Exterior Angle Theorem and perpendiculars
Triangle congruence conditions
Three inequalities for triangles
The Alternate Interior Angles Theorem
The Saccheri-Legendre Theorem
Quadrilaterals
Statements equivalent to the Euclidean Parallel Postulate
Rectangles and defect
The Universal Hyperbolic Theorem
Euclidean Geometry
Basic theorems of Euclidean geometry
The Parallel Projection Theorem
Similar triangles
The Pythagorean Theorem
Trigonometry
Exploring the Euclidean geometry of the triangle
Hyperbolic Geometry
The discovery of hyperbolic geometry
Basic theorems of hyperbolic geometry
Common perpendiculars
Limiting parallel rays and asymptotically parallel lines
Properties of the critical function
The defect of a triangle
Is the real world hyperbolic?
Area
The Neutral Area Postulate
Area in Euclidean geometry
Dissection theory in neutral geometry
Dissection theory in Euclidean geometry
Area and defect in hyperbolic geometry
Circles
Basic definitions
Circles and lines
Circles and triangles
Circles in Euclidean geometry
Circular continuity
Circumference and area of Euclidean circles
Exploring Euclidean circles
Constructions
Compass and straightedge constructions
Neutral constructions
Euclidean constructions
Construction of regular polygons
Area constructions
Three impossible constructions
Transformations
The transformational perspective
Properties of isometries
Rotations, translations, and glide reflections
Classification of Euclidean motions
Classification of hyperbolic motions
Similarity transformations in Euclidean geometry
A transformational approach to the foundations
Euclidean inversions in circles
Models
The significance of models for hyperbolic geometry
The Cartesian model for Euclidean geometry
The Poincar� disk model for hyperbolic geometry
Other models for hyperbolic geometry
Models for elliptic geometry
Regular Tessellations
Polygonal Models and the Geometry of Space
Curved surfaces
Approximate models for the hyperbolic plane
Geometric surfaces
The geometry of the universe
Conclusion
Further study
Templates
Appendices
Euclid's Book I
Definitions
Postulates
Common Notions
Propositions
Systems of Axioms for Geometry
Filling in Euclid's gaps
Hilbert's axioms
Birkhoff's axioms
MacLane's axioms
SMSG axioms
UCSMP axioms
The Postulates Used in this Book
The undefined terms
Neutral postulates
Parallel postulates
Area postulates
The reflection postulate
Logical relationships
Set Notation and the Real Numbers
Some elementary set theory
Properties of the real numbers
Functions
The van Hiele Model
Hints for Selected Exercises
Bibliography
Index

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