AMSCO'S Geometry. New York - Rye High School

Points, Lines, and Planes 421Theorem 11.2If two lines intersect, then there is exactly one plane containing them.This theorem can be stated in another way:Theorem 11.2Two intersecting lines determine a plane.Given Lines l and m intersecting at point P.ProveThere is exactly one plane containing land m.PAm BProof Choose two points, A on line l and B on line m. The three points, A, B, and P,determine one and only one plane. A plane containing any two points containsall of the points on the line determined by those two points. Since the two pointsA and P on line l are on the plane, then all of the points of l are on the plane.Since the two points B and P on line m are on the plane, then all of the pointsof m are on the plane. Therefore, there is exactly one plane that contains thegiven intersecting lines.The definition of parallel lines gives us another set of points that must lie ina plane.DEFINITIONParallel lines in space are lines in the same plane that have no points in common.lThis definition can be written as a biconditional:Two lines are parallel if andonly if they are coplanar and have no points in common. If AB gand CD gare parallellines in space, then they determine a plane if and only if they are two distinctlines.We have seen that intersecting lines lie ina plane and parallel lines lie in a plane. Forexample, in the diagram, AB g CD gso AB gandHGCD glie in a plane. Also, AB gand BF gare intersectinglines so they lie in a plane. But thereare some pairs of lines that do not intersectDCand are not parallel. In the diagram, AB gandECG gFare neither intersecting nor parallel lines.AB g and CG gare called skew lines. BF gand EH gABare another pair of skew lines.