AMSCO'S Geometry. New York - Rye High School

438 The Geometry of Three DimensionsTheorem 11.13Parallel planes are everywhere equidistant.Given Parallel planes p and q, with AC and BD eachperpendicular to p and q with an endpoint oneach plane.pProveProofAC BDTwo lines perpendicular to the same plane areboth parallel and coplanar. Therefore, AC BDand lie on the same plane. That plane intersectsparallel planes p and q in parallel lines AB g andCD g . In the plane of AC gand BD g, ABDC is aparallelogram with a right angle, that is, a rectangle.Therefore, AC and BD are congruent andAC BD.ACDBqEXAMPLE 2Line l is perpendicular to plane p and linel is not perpendicular to plane q. Is p q?SolutionAssume that p q. If two planes are parallel,then a line perpendicular to one isperpendicular to the other. Therefore,since l is perpendicular to plane p, l mustbe perpendicular to plane q. This contradictsthe given statement that l is not perpendicularto q, and the assumption isfalse. Therefore, p is not parallel to q.qplEXAMPLE 3Given: AB g ' plane p at A, CD g ' plane p at C,and AB CD.BDProve: A, B, C, and D are the vertices of a parallelogram.ACp