AMSCO'S Geometry. New York - Rye High School

434 The Geometry of Three DimensionsEXAMPLE 1Plane p intersects plane q in AB gandplane r in CD g. Prove that if AB gandCD gintersect, then planes q and r arenot parallel.AqCrProof Let E be the point at which AB gandCD gintersect. Then E is a point on qand E is a point on r. Therefore, planesq and r intersect in at least one pointand are not parallel.BDEpTheorem 11.10If a plane intersects two parallel planes, then the intersection is two parallellines.Given Plane p intersects plane m at AB gand plane n atCD g, m n.Prove AB g CD gAmpBProofUse an indirect proof.Lines AB gand CD gare two lines of plane p.Twolines in the same plane either intersect or are parallel.n CDIf AB gis not parallel to CD gthen they intersect in some point E. Since Eis a point of AB g, then it is a point of plane m. Since E is a point of CD g, then itis a point of plane n. But m n and have no points in common. Therefore, AB gand CD gare two lines in the same plane that do not intersect, and AB g CD g.In a plane, two lines perpendicular to a given line are parallel. Can we provethat two lines perpendicular to a given plane are also parallel?