Plane Geometry - Bruce E. Shapiro

Section 11The UCSMP AxiomsThe University of Chicago School MathematicsProject was founded in 1983 with the aimof upgrading mathematics education in elementaryand secondary schools throughoutthe United States. They have developed a setof axioms that are in wide use today, and arealso redundant in the sense that some axiomscan be proved from others. The purpose of theredundancy was to make the learning of geometrymore intuitive. These axioms used incorporateda transformational approach. Detailsof the projects history is given on itsweb page at (http://ucsmp.uchicago.edu/history.html). Many of today’s elementaryand secondary textbooks are based on these standards, which encompassall of mathematics, not just geometry.The only undefined terms are point, line, and plane.Point-Line-Plane AxiomsAxiom 1 Through any two points there is exactly one line.Axiom 2 Every line is a set of points that can be put into a one-to-onecorrespondence with the real numbers, with any point corresponding tozero and any other point corresponding to the number 1.Axiom 3 Given a line in a plane, there is at least one point in the planethat is not on the line. Given a plane in space, there is at least one point45