Foundations of Geometry

by Venema, Gerard A.

• ISBN13: 9780131437005
• ISBN10: 0131437003
• Edition: 1st
• Format: Paperback
• Copyright: 2005-01-01
• Publisher: Prentice Hall

Summary

For sophomore/junior-level courses in Geometry; especially appropriate for students that will go on to teach high-school mathematics.This text comfortably serves as a bridge between lower-level mathematics courses (calculus and linear algebra) and upper-level courses (real analysis and abstract algebra). It fully implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers. Foundations of Geometry particularly teaches good proof-writing skills, emphasizes the historical development of geometry, and addresses certain issues concerning the place of geometry in human culture.

Table of Contents

Preface	ix
1 Euclid's Elements	1	(16)
1.1 Geometry before Euclid	1	(1)
1.2 The logical structure of Euclid's Elements	2	(1)
1.3 The historical importance of Euclid's Elements	3	(2)
1.4 A look at Book I of the Elements	5	(3)
1.5 A critique of Euclid's Elements	8	(3)
1.6 Final observations about the Elements	11	(6)
2 Axiomatic Systems and Incidence Geometry	17	(14)
2.1 Undefined and defined terms	17	(1)
2.2 Axioms	18	(1)
2.3 Theorems	18	(1)
2.4 Models	19	(1)
2.5 An example of an axiomatic system	19	(6)
2.6 The parallel postulates	25	(2)
2.7 Axiomatic systems and the real world	27	(4)
3 Theorems, Proofs, and Logic	31	(12)
3.1 The place of proof in mathematics	31	(1)
3.2 Mathematical language	32	(2)
3.3 Stating theorems	34	(3)
3.4 Writing proofs	37	(2)
3.5 Indirect proof	39	(1)
3.6 The theorems of incidence geometry	40	(3)
4 Set Notation and the Real Numbers	43	(9)
4.1 Some elementary set theory	43	(2)
4.2 Properties of the real numbers	45	(3)
4.3 Functions	48	(1)
4.4 The foundations of mathematics	49	(3)
5 The Axioms of Plane Geometry	52	(42)
5.1 Systems of axioms for geometry	53	(3)
5.2 The undefined terms	56	(1)
5.3 Existence and incidence	56	(1)
5.4 Distance	57	(6)
5.5 Plane separation	63	(4)
5.6 Angle measure	67	(3)
5.7 Betweenness and the Crossbar Theorem	70	(14)
5.8 Side-Angle-Side	84	(5)
5.9 The parallel postulates	89	(1)
5.10 Models	90	(4)
6 Neutral Geometry	94	(41)
6.1 Geometry without the parallel postulate	94	(1)
6.2 Angle-Side-Angle and its consequences	95	(2)
6.3 The Exterior Angle Theorem	97	(5)
6.4 Three inequalities for triangles	102	(5)
6.5 The Alternate Interior Angles Theorem	107	(3)
6.6 The Saccheri-Legendre Theorem	110	(3)
6.7 Quadrilaterals	113	(3)
6.8 Statements equivalent to the Euclidean Parallel Postulate	116	(7)
6.9 Rectangles and defect	123	(8)
6.10 The Universal Hyperbolic Theorem	131	(4)
7 Euclidean Geometry	135	(26)
7.1 Geometry with the parallel postulate	135	(2)
7.2 Basic theorems of Euclidean geometry	137	(2)
7.3 The Parallel Projection Theorem	139	(2)
7.4 Similar triangles	141	(2)
7.5 The Pythagorean Theorem	143	(2)
7.6 Trigonometry	145	(2)
7.7 Exploring the Euclidean geometry of the triangle	147	(14)
8 Hyperbolic Geometry	161	(33)
8.1 The discovery of hyperbolic geometry	161	(2)
8.2 Basic theorems of hyperbolic geometry	163	(5)
8.3 Common perpendiculars	168	(3)
8.4 Limiting parallel rays and asymptotically parallel lines	171	(10)
8.5 Properties of the critical function	181	(4)
8.6 The defect of a triangle	185	(4)
8.7 Is the real world hyperbolic?	189	(5)
9 Area	194	(31)
9.1 The Neutral Area Postulate	195	(3)
9.2 Area in Euclidean geometry	198	(8)
9.3 Dissection theory in neutral geometry	206	(7)
9.4 Dissection theory in Euclidean geometry	213	(3)
9.5 Area and defect in hyperbolic geometry	216	(9)
10 Circles	225	(39)
10.1 Basic definitions	226	(1)
10.2 Circles and lines	227	(4)
10.3 Circles and triangles	231	(7)
10.4 Circles in Euclidean geometry	238	(6)
10.5 Circular continuity	244	(3)
10.6 Circumference and area of Euclidean circles	247	(8)
10.7 Exploring Euclidean circles	255	(9)
11 Constructions	264	(20)
11.1 Compass and straightedge constructions	265	(2)
11.2 Neutral constructions	267	(3)
11.3 Euclidean constructions	270	(2)
11.4 Construction of regular polygons	272	(4)
11.5 Area constructions	276	(3)
11.6 Three impossible constructions	279	(5)
12 Transformations	284	(43)
12.1 The transformational perspective	285	(1)
12.2 Properties of isometrics	286	(6)
12.3 Rotations, translations, and glide reflections	292	(8)
12.4 Classification of Euclidean motions	300	(3)
12.5 Classification of hyperbolic motions	303	(1)
12.6 A transformational approach to the foundations	304	(6)
12.7 Euclidean inversions in circles	310	(17)
13 Models	327	(18)
13.1 The significance of models for hyperbolic geometry	327	(2)
13.2 The Cartesian model for Euclidean geometry	329	(2)
13.3 The Poincaré disk model for hyperbolic geometry	331	(5)
13.4 Other models for hyperbolic geometry	336	(5)
13.5 Models for elliptic geometry	341	(4)
14 Polygonal Models and the Geometry of Space	345	(47)
14.1 Curved surfaces	346	(11)
14.2 Approximate models for the hyperbolic plane	357	(6)
14.3 Geometric surfaces	363	(6)
14.4 The geometry of the universe	369	(6)
14.5 Conclusion	375	(1)
14.6 Further study	375	(9)
14.7 Templates	384	(8)
APPENDICES
A Euclid's Book I	392	(6)
A.1 Definitions	392	(2)
A.2 Postulates	394	(1)
A.3 Common Notions	394	(1)
A.4 Propositions	394	(4)
B Other Axiom Systems	398	(10)
B.1 Hilbert's axioms	398	(2)
B.2 Birkhoff's axioms	400	(1)
B.3 MacLane's axioms	401	(1)
B.4 SMSG axioms	402	(3)
B.5 UCSMP axioms	405	(3)
C The Postulates Used in this Book	408	(3)
C.1 The undefined terms	408	(1)
C.2 The postulates of neutral geometry	408	(1)
C.3 The parallel postulates	409	(1)
C.4 The area postulates	409	(1)
C.5 The reflection postulate	410	(1)
C.6 Logical relationships	410	(1)
D The van Hiele Model	411	(1)
E Hints for Selected Exercises	412	(9)
Bibliography	421	(4)
Index	425