AMSCO'S Geometry. New York - Rye High School

420 The Geometry of Three Dimensions11-1 POINTS, LINES, AND PLANESIn this text, we have been studying points and lines in a plane, that is, the geometryof two dimensions. But the world around us is three-dimensional. Thegeometry of three dimensions is called solid geometry. We begin this study withsome postulates that we can accept as true based on our observations.We know that two points determine a line. How many points are needed todetermine a plane? A table or chair that has four legs will sometimes beunsteady on a flat surface. But a tripod or a stool with three legs always sitsfirmly. This observation suggests the following postulate.Postulate 11.1There is one and only one plane containing three non-collinear points.For a set of the three non-collinearpoints that determine a plane, eachpair of points determines a line and allof the points on that line are points ofthe plane.ACBPostulate 11.2A plane containing any two points contains all of the points on the linedetermined by those two points.These two postulates make it possible for us to prove the followingtheorems.Theorem 11.1There is exactly one plane containing a line and a point not on the line.Given Line l and point P not on l.ProveThere is exactly one plane containingl and P.ABlPProofChoose two points A and B on line l. The three points, A, B, and P, determineone and only one plane. If the two points A and B on line l are on the plane, thenall of the points of l are on the plane, that is, the plane contains line l. Therefore,there is exactly one plane that contains the given line and point.