Plane Geometry - Bruce E. Shapiro

42 SECTION 10. THE SMSG AXIOMSP is zero and the coordinate of Q is positive.Axiom 5 Every plane contains at least three non-collinear points, andspace contains at least four non-coplanar points.Axiom 6 If two points lie in a plane, then the line containing these pointslies in the same plane.Axiom 7 Any three points lie in at least one plane, and any three noncollinearpoints lie in exactly one plane.Axiom 8 If two planes intersect, then that intersection is a line.Edward Griffith Begle (1941-1978), director of SMSG for ten years. Photograph by PaulHalmos, Archives of American Mathematics, Dolph Briscoe Center for American History,University of Texas at Austin.Axiom 9 Plane Separation Postulate. Given a line and a plane containingit, the points of the plane that do not lie on the line form two sets suchthat: (1) each of the sets is convex; and (2) if P is in one set and Q is inthe other, then segment PQ intersects the line.Axiom 10 Space Separation Postulate. The points of space that do not liein a given plane form two sets such that: (1) Each of the sets is convex;and (2) If P is in one set and Q is in the other, then segment PQ intersectsthe plane.Axiom 11 Angle Measurement Postulate. To every angle ∠x there correspondsa real number between 0 ◦ and 180 ◦ .The real number is called the measure of the angle and denoted by m(∠x).Axiom 12 Angle Construction postulate. Let AB be a ray on the edge ofthe half-plane H. For every r between 0 and 180 there is exactly one ray« CC BY-NC-ND 3.0. Revised: 18 Nov 2012