
Foundations of Plane Geometry, 1/e
Harvey I. Blau, Northern Illinois University Coming February, 2000 by Prentice Hall Engineering/Science/Mathematics
 
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—Defines concepts as soon as they make sense in the general development, but not necessarily before all axioms which relate to them have been introduced. —This often leads to some thoughtprovoking examples—e.g., when “segment” is defined in Chapter 6, an example (the “InsideOut” model) and a result (Proposition 6.4) show that at this stage, “length” is an invariant of a segment but “endpoints” are not. Unique approach to the standard sets of axioms—Facilitates incremental learning, e.g.: —The “Betweenness” and “Quadrichotomy” axioms for points are simply stated assumptions which permit the development of much of the theory of betweenness for collinear points even though there are many nonstandard (even finite) models which satisfy these postulates. —The relatively natural and terse “Real Ray Axiom” allows the completion of the theory of betweenness of points. —Most of the axioms for coterminal rays are formulated as exact analogs of previous axioms for collinear points. This analogy (“duality”) is invoked to instantly establish many properties of coterminal rays. The articulation of these properties reinforces understanding of the previous results about collinear points. Original treatments of most topics—e.g.: —The only polygons used in the development are triangles. The notions of rectangles and Saccheri quadrilaterals are confined to exercises. This leads to a novel proof of the theorem on angle sums of triangles in spherical planes (18.1). —The separate introduction of halfplanes in Chapter 10 and fans of coterminal rays in Chapter 11 and the proof of their relationship in Theorem 12.2. The Crossbar Theorem (12.4) is then an almost immediate consequence. States and discusses the Ruler and Protractor Axioms (commonly used in secondary school geometry texts) as theorems—After all the axioms have been introduced (Chapter 13) — and gives their proofs as structured exercises. Careful but not overly wordy explanations throughout the text. —All proofs are written in detail in paragraph style. —Informal explanations—and references to specific models—are interspersed frequently. —Figures illustrate virtually all of the definitions and the key points of most proofs. Informal chapter on logic (Ch. 2)—Addresses students' most common blunders and misconceptions and introduces basic ideas about proofs.
—Some call for proofs of results which are stated in the book without proof; some are fairly immediate applications of the theorems in the text; some involve longer chains of reasoning and often include hints; and others develop concepts which are omitted from the text such as Saccheri quadrilaterals. —A simple type of exercise—effective in developing comprehension and articulation—asks students to rewrite the statement of a particular axiom or theorem entirely in words without the use of symbols or lettersassymbols.
1. Five Examples. 2. Some Logic. 3. Practice Proofs. 4. Set Terminology and Sets of Real Numbers. 5. An Axiom System for Plane Geometry: First Steps. 6. Betweenness, Segments and Rays. 7. Three Axioms for the Line. 8. The Real Ray Axiom and Its Consequences. 9. Antipodes and Opposite Rays. 10. Separation. 11. Pencils and Angles. 12. The Crossbar Theorem. 13. SideAngleSide. 14. Perpendiculars. 15. The Exterior Angle Inequality and Triangle Inequality. 16. Further Inequalities Concerning Triangles. 17. Parallels and the Diameter of the Plane. 18. Angle Sums of Triangles. 19. Parallels and Angle Sums. 20. Concurrence Theorems. 21. Circles. 22. Similarity. Bibliography. Appendix I. Definitions and Assumptions from Book I of Euclid's Elements. Appendix II. The SideAngleSide Axiom in the Hyperbolic Plane. Index.
