Plane Geometry - Bruce E. Shapiro

Section 10The SMSG AxiomsThe School Mathematics Study Group (SMSG) at Yale University wasfunded by the US National Science Foundation to reform mathematics educationin the 1950’s and developed mathematical curricula that came beknow as the “new math” during the 1960’s. A set of 22 s that were intendedto make geometry more intuitive and understandable were produced by thisgroup, as was a mimeographed textbook that was later used as the basis ofa geometry textbook by E.E. Moise and F.L.Downs (1964) that is still incirculation. Some of the axioms are redundant in the sense that they canbe derived from the others.The undefined terms are point, line, plane, lie on, distance, angle measure,area, volume, and there are 22 axioms.Axiom 1 Given any two distinct points there is exactly one line that containsthem.Axiom 2 Distance Postulate. To every pair of distinct points there correspondsa unique positive number. This number is called the distancebetween the two points.Axiom 3 Ruler Postulate. The points of a line can be placed in a correspondencewith the real numbers such that: (1) To every point of the linethere corresponds exactly one real number; (2) To every real number therecorresponds exactly one point of the line. (3) The distance between twodistinct points is the absolute value of the difference of the correspondingreal numbers.Axiom 4 Ruler Placement Postulate. Given two points P and Q of a line,the coordinate system can be chosen in such a way that the coordinate of41