parallel and perpendicular planes

Theorem 1.  If a plane ($\pi$) intersects two parallel planes ($\varrho$, $\sigma$), the intersection lines are parallel.

Proof.  The intersection lines cannot have common points, because $\varrho$ and $\sigma$ have no such ones.  Since the lines are in a same plane $\pi$, they are parallel.

Theorem 2.  If a plane ($\pi$) contains the normal (http://planetmath.org/PlaneNormal) ($n$) of another plane ($\varrho$), the planes are perpendicular (http://planetmath.org/DihedralAngle) to each other.

Proof.  Draw in the plane $\varrho$ the line $l$ cutting the intersection line perpendicularly and cutting also $n$.  Then $l$ must be perpendicular to $n$ and thus to the whole plane $\pi$ (see the Theorem in the entry normal of plane).  Consequently, the right angle formed by the lines $n$ and $l$ is the normal section of the dihedral angle formed by the planes $\pi$ and $\varrho$.  Therefore,  $\pi\bot\varrho$.

Title parallel and perpendicular planes ParallelAndPerpendicularPlanes 2013-04-19 15:18:51 2013-04-19 15:18:51 pahio (2872) pahio (2872) 10 pahio (2872) Theorem msc 51M04 PlaneNormal NormalOfPlane ParallelismOfTwoPlanes