# Foundations of Geometry, 3rd edition

Published by Pearson (April 15, 2021) © 2022

**Gerard A. Venema**

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For courses in Advanced Geometry or Geometry for Future Elementary and Middle School Teachers.

### Experience the beauty and fascination of geometry

**Foundations of Geometry, 3rd Edition **enriches the education of all mathematics majors and encourages a smooth transition into more advanced courses. Venema 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.

### Hallmark features of this title

**Achieves****the goals of MET and CCSM**in the context of a traditional axiomatic approach to geometry.**Comprehensive coverage of most of Euclid's****Elements**includes almost all the material in the first 6 books of that work.**Careful statements of the axioms**help students understand how the theorems are built on 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.**An introduction to proof**bridges lower-level courses (in which technique is emphasized) to upper-level courses (in which proof and the understanding of concepts are emphasized).

### New and updated features of this title

**For the first time, this revision is available as a Pearson eText.**The eText includes GeoGebra widgets that allow users to explore various constructions and to make discoveries for themselves.**More exercises**have been added throughout.**Material has been refined and rewritten**as needed to bring in line with current standards.**User feedback**from the previous edition informs the presentation of the material in several sections throughout.

### Features of Pearson eText for the 3rd Edition

**For the first time, this revision is available as a Pearson eText.**The eText includes GeoGebra widgets that allow users to explore various constructions and to make discoveries for themselves.

**1. Prologue: Euclid's Elements**

1.1 Geometry before Euclid

1.2 The Logical Structure of Euclid's Elements

1.3 The Historical Significance of Euclid's Elements

1.4 A Look at Book I of the Elements

1.5 A Critique of Euclid's Elements

1.6 A New View of the Foundations

1.7 Some Final Observations about the Elements

**2. Axiomatic Systems and Incidence Geometry**

2.1 The Structure of an Axiomatic System

2.2 An Example: Incidence Geometry

2.3 The Parallel Postulates in Incidence Geometry

2.4 Axiomatic Systems and the RealWorld

2.5 Theorems, Proofs, and Logic

2.6 Some Theorems from Incidence Geometry

**3. A System of Axioms for Plane Geometry**

3.1 The Undefined Terms and Two Fundamental Axioms

3.2 Distance and the Ruler Postulate

3.3 The Plane Separation Postulate

3.4 Angle Measure and the Protractor Postulate

3.5 The Crossbar Theorem and the Linear Pair Theorem

3.6 The Side-Angle-Side Postulate

3.7 The Parallel Postulates and Models for Neutral Geometry

**4. Neutral Geometry**

4.1 The Exterior Angle Theorem

4.2 Triangle Congruence Conditions

4.3 Three Inequalities for Triangles

4.4 The Alternate Interior Angles Theorem

4.5 The Saccheri-Legendre Theorem

4.6 Quadrilaterals

4.7 Statements Equivalent to the Euclidean Parallel Postulate

4.8 Rectangles and Defect

4.9 The Universal Hyperbolic Theorem

**5. Euclidean Geometry**

5.1 Basic Theorems of Euclidean Geometry

5.2 The Parallel Projection Theorem

5.3 Similar Triangles

5.4 The Pythagorean Theorem

5.5 Trigonometry

5.6 Exploring the Geometry of Euclidean Triangles

**6. Hyperbolic Geometry**

6.1 Basic Theorems of Hyperbolic Geometry

6.2 Common Perpendiculars

6.3 The Angle of Parallelism

6.4 Limiting Parallel Rays

6.5 Asymptotic Triangles

6.6 The Classification of Parallels

6.7 The Critical Function

6.8 The Defect of a Triangle

6.9 Is the Real World Hyperbolic?

**7. Area**

7.1 The 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 The Associated Saccheri Quadrilateral

7.6 Area and Defect in Hyperbolic Geometry

7.7 The Euclidean Area Postulate Reconsidered

**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 in Geometry

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 Isometries of the Plane

10.2 Rotations, Translations, and Glide Reflections

10.3 Classification of Euclidean Motions

10.4 Classification of Hyperbolic Motions

10.5 A Transformational Approach to the Foundations

10.6 Similarity Transformations in Euclidean Geometry

10.7 Euclidean Inversions in Circles

**11. Models**

11.1 The Cartesian Model for Euclidean Geometry

11.2 The Poincaré 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 The Geometry of the Universe

12.5 Conclusion

12.6 Further Study

12.7 Templates

**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 Birkhoff's Axioms

B.3MacLane'sAxioms

B.4 SMSG Axioms

B.5 UCSMP Axioms

**C. The Postulates Used in This Book**

C.1 Criteria Used in Selecting the Postulates

C.2 Statements of the Postulates

C.3 Logical Relationships

**D. The Van Hiele Model of the Development of Geometric Thought**

**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

**Gerard Venema **earned an A.B. in mathematics from Calvin College and a Ph.D. from the University of Utah. After completing his education, he spent two years in a postdoctoral position at the University of Texas at Austin and another two years as a Member of the Institute for Advanced Study in Princeton, NJ. He then returned to his alma mater, Calvin University, and has been a faculty member there ever since. While on the Calvin University faculty he also held visiting faculty positions at the University of Tennessee, the University of Michigan, and Michigan State University. He also spent two years as Program Director for Topology, Geometry, and Foundations in the Division of Mathematical Sciences at the National Science Foundation and nearly ten years as the Associate Secretary of the Mathematical Association of America.

Venema is a member of the American Mathematical Society and the Mathematical Association of America. He is the author of two other books. One is an undergraduate textbook, ** Exploring Advanced Euclidean Geometry with GeoGebra**, published by the Mathematical Association of America. The other is a research monograph,

**, coauthored by Robert J. Daverman, that was published by the American Mathematical Society as volume 106 in its Graduate Studies in Mathematics series. In addition, Venema is author of over thirty research articles in geometric topology.**

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