# 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 parallel to a line ($m$) contained in a plane ($\pi$), then it is parallel to the plane or is contained in the plane.

Proof.  So,  $l\,||\,m\subset\pi$.  If  $l\not\subset\pi$,  we can set a set along the parallel lines $l$ and $m$ another plane $\varrho$.  The common points of $\pi$ and $\varrho$ are on the intersection line $m$ of the planes.  If $l$ would intersect the plane $\pi$, then it would intersect also the line $m$, contrary to the assumption.  Thus  $l\,||\,\pi$.

Theorem 2.  If a plane is set along a line ($l$) which is parallel to another plane ($\pi$), 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 $\pi$ which would contradict the assumption.  Accordingly,  $m\,||\,l$.

Title parallelism of line and plane ParallelismOfLineAndPlane 2013-03-22 18:47:58 2013-03-22 18:47:58 pahio (2872) pahio (2872) 6 pahio (2872) Theorem msc 51M04 ParallelismOfTwoPlanes