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2. Relationship between neutral geometry and incidence geometry. (a) Explain how the axioms stated in this chapter imply that there exists at least one line. (b) Explain how the existence of three noncollinear points (Incidence Axiom I-3) can be deduced from the axioms stated in this chapter. (c) Explain why every model for neutral geometry must have infinitely many points. (d) Explain why every model for neutral geometry must have infinitely many lines. (e) Explain why every model for neutral geometry must also be a model for incidence geometry. (f) Explain why all the theorems of incidence geometry must also be theorems in neutral geometry.

2. Relationship between neutral geometry and incidence geometry. (a) Explain how the axioms stated in this chapter imply that there exists at least one line. (b) Explain how the existence of three noncollinear points (Incidence Axiom I-3) can be deduced from the axioms stated in this chapter. (c) Explain why every model for neutral geometry must have infinitely many points. (d) Explain why every model for neutral geometry must have infinitely many lines. (e) Explain why every model for neutral geometry must also be a model for incidence geometry. (f) Explain why all the theorems of incidence geometry must also be theorems in neutral geometry.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.2: Statements And Reasoning
Problem 56E: In Exercises 55 and 56, P is a true statement, while Q and R are false statements. Classify each of...
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proved that the axioms of neutral geometry are consistent. For now we will proceed in the faith that there is a model and will revisit the question of consistency in a later chapter. Even though we will treat the three geometries axiomatically, we will still want to draw diagrams that illustrate the relationships being discussed. A diagram should be viewed as one possible interpretation of a theorem. Both the theorem statements and the proofs should stand alone and should not depend on the diagrams. The purpose of a diagram is to help us to visually organize the information being discussed. Diagrams add tremendously to our intuitive understanding, but they are only meant to illustrate, not to narrow the range of possible interpretations. EXERCISES 3.7 1. Check that the trivial geometry containing just one point and no lines satisfies all the postulates of neutral geometry except the Existence Postulate. Which parallel postulate 68 Chapter 3 Axioms for Plane Geometry is satisfied by this geometry? (This exercise is supposed to convince you of the need for the Existence Postulate.) 2. Relationship between neutral geometry and incidence geometry. (a) Explain how the axioms stated in this chapter imply that there exists at least one line. (b) Explain how the existence of three noncollinear points (Incidence Axiom I-3) can be deduced from the axioms stated in this chapter. (c) Explain why every model for neutral geometry must have infinitely many points. (d) Explain why every model for neutral geometry must have infinitely many lines. (e) Explain why every model for neutral geometry must also be a model for incidence geometry. (f) Explain why all the theorems of incidence geometry must also be theorems in neutral geometry. СНАРТER 4 Neutral Geometry 4.1 THE EXTERIOR ANGLE THEOREM AND EXISTENCE OF PERPENDICULARS 4.2 TRIANGLE CONGRUENCE CONDITIONS 4.3 THREE INEQUALITIES FOR TRIANGLES 4.4 THE ALTERNATE INTERIOR ANGLES THEOREM
Transcribed Image Text:proved that the axioms of neutral geometry are consistent. For now we will proceed in the faith that there is a model and will revisit the question of consistency in a later chapter. Even though we will treat the three geometries axiomatically, we will still want to draw diagrams that illustrate the relationships being discussed. A diagram should be viewed as one possible interpretation of a theorem. Both the theorem statements and the proofs should stand alone and should not depend on the diagrams. The purpose of a diagram is to help us to visually organize the information being discussed. Diagrams add tremendously to our intuitive understanding, but they are only meant to illustrate, not to narrow the range of possible interpretations. EXERCISES 3.7 1. Check that the trivial geometry containing just one point and no lines satisfies all the postulates of neutral geometry except the Existence Postulate. Which parallel postulate 68 Chapter 3 Axioms for Plane Geometry is satisfied by this geometry? (This exercise is supposed to convince you of the need for the Existence Postulate.) 2. Relationship between neutral geometry and incidence geometry. (a) Explain how the axioms stated in this chapter imply that there exists at least one line. (b) Explain how the existence of three noncollinear points (Incidence Axiom I-3) can be deduced from the axioms stated in this chapter. (c) Explain why every model for neutral geometry must have infinitely many points. (d) Explain why every model for neutral geometry must have infinitely many lines. (e) Explain why every model for neutral geometry must also be a model for incidence geometry. (f) Explain why all the theorems of incidence geometry must also be theorems in neutral geometry. СНАРТER 4 Neutral Geometry 4.1 THE EXTERIOR ANGLE THEOREM AND EXISTENCE OF PERPENDICULARS 4.2 TRIANGLE CONGRUENCE CONDITIONS 4.3 THREE INEQUALITIES FOR TRIANGLES 4.4 THE ALTERNATE INTERIOR ANGLES THEOREM
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