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单词 ParallelismOfLineAndPlane
释义

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.

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更新时间:2025/5/4 20:28:21