parallelism of line and plane

Parallelity of a line and a plane means that the angle between line and plane is 0, i.e. the line and the plane have either no or infinitely many common points.

Theorem 1.  If a line (l) is parallelMathworldPlanetmathPlanetmath to a line (m) contained in a plane (π), then it is parallel to the plane or is contained in the plane.

Proof.  So,  l||mπ.  If  lπ,  we can set a set along the parallel lines l and m another plane ϱ.  The common points of π and ϱ are on the intersectionMathworldPlanetmath line m of the planes.  If l would intersect the plane π, then it would intersect also the line m, contrary to the assumptionPlanetmathPlanetmath.  Thus  l||π.

Theorem 2.  If a plane is set along a line (l) which is parallel to another plane (π), then the intersection line (m) of the planes is parallel to the first-mentioned line.

Proof.  The lines l and m are in a same plane, and they cannot intersect each other since otherwise l would intersect the plane π which would contradict the assumption.  Accordingly,  m||l.

Title parallelism of line and plane
Canonical name ParallelismOfLineAndPlane
Date of creation 2013-03-22 18:47:58
Last modified on 2013-03-22 18:47:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 51M04
Related topic ParallelismOfTwoPlanes