Plane Geometry - Bruce E. Shapiro

Section 12Venema’s AxiomsVenema [Venema, 2006] introduces a set of axioms that meld together partsof the axiom systems of Birkhoff, MacLane, the SMSG, and the UCSMP.We present them here for reference, since we will be using this system inthe remainder of the class.Venema’s Undefined TermsThe undefined terms are point, line, distance, half-plane, angle-measure,areaVenema’s Axiom’s of Neutral GeometryExistence says that at least some points exist, and incidents says that everypair of distinct points defines a line.1. Existence Postulate. The collection of all points forms a nonemptyset with more than one (i.e., at least two) points. The set of all points inthe plane is called P.2. Incidence Postulate. Every line is a set of points. For every pair ofdistinct points A, B there is exactly one line l = ←→ AB such that A, B ∈ l.The ruler postulate allows us to associate real numbers – and hence measurements– with distances and line segments.49