Question 2 proved that the axioms of neutral geometry are consistent. For now we will proceed in thefaith 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 todraw diagrams that illustrate the relationships being discussed. A diagram should beviewed as one possible interpretation of a theorem. Both the theorem statements andthe proofs should stand alone and should not depend on the diagrams. The purpose of adiagram is to help us to visually organize the information being discussed. Diagrams addtremendously to our intuitive understanding, but they are only meant to illustrate, not tonarrow the range of possible interpretations.EXERCISES 3.71. Check that the trivial geometry containing just one point and no lines satisfies all thepostulates of neutral geometry except the Existence Postulate. Which parallel postulate68Chapter 3Axioms for Plane Geometryis satisfied by this geometry? (This exercise is supposed to convince you of the need forthe 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 oneline.(b) Explain how the existence of three noncollinear points (Incidence Axiom I-3) canbe 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 incidencegeometry.(f) Explain why all the theorems of incidence geometry must also be theorems inneutral geometry.СНАРТER4Neutral Geometry4.1THE EXTERIOR ANGLE THEOREM AND EXISTENCE OF PERPENDICULARS4.2 TRIANGLE CONGRUENCE CONDITIONS4.3THREE INEQUALITIES FOR TRIANGLES4.4THE ALTERNATE INTERIOR ANGLES THEOREM

note :As per the instructions we will only solve part (2) of the question.In neutral geometry, there…

Answered: 2. Relationship between neutral…

Linear Algebra: A Modern Introduction

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Chapter1: Vectors

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