Foundations of Geometry by Gerard A. Venema

By Gerard A. Venema

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8. Find the first of Euclid’s proofs in which he makes use of his Fifth Postulate. 9. A rhombus is a quadrilateral in which all four sides have equal lengths. The diagonals are the line segments joining opposite corners. Use the first five Propositions of Book I of the Elements to show that the diagonals of a rhombus divide the rhombus into four congruent triangles. 10. A rectangle is a quadrilateral in which all four angles have equal measures. ) Use the propositions in Book I of the Elements to show that the diagonals of a rectangle are congruent and bisect each other.

We do not apply the terms true or false to the axioms in any absolute sense. An axiom is simply a statement that may be true or may be false in any particular situation, it just depends on how we choose to interpret the undefined terms. Thus our efforts to introduce abstraction and rigor into geometry have led us to drain the meaning out of such everyday terms as point and line. Since the words can now mean just about anything we want them to, we must wonder whether they any longer have any real content.

Rather they are additional statements that may or may not be satisfied by a particular model for incidence geometry. We illustrate with several examples. 2 Parallelism in three-, four-, and five-point geometries In three-point geometry any two lines intersect. Therefore there are no parallel lines and this model satisfies the Elliptic Parallel Postulate. Each line in four-point geometry is disjoint from exactly one other line. Thus, for example, the line {A, B} is parallel to the line {C, D} and no others.

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