parallel and perpendicular planes

Theorem 1.  If a plane (π) intersects two parallel planesMathworldPlanetmath (ϱ, σ), the intersection lines are parallelMathworldPlanetmathPlanetmath.

Proof.  The intersection lines cannot have common points, because ϱ and σ have no such ones.  Since the lines are in a same plane π, they are parallel.

Theorem 2.  If a plane (π) contains the normal ( (n) of another plane (ϱ), the planes are perpendicularMathworldPlanetmathPlanetmathPlanetmath ( to each other.

Proof.  Draw in the plane ϱ 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 π (see the Theorem in the entry normal of plane).  Consequently, the right angleMathworldPlanetmathPlanetmath formed by the lines n and l is the normal section of the dihedral angle formed by the planes π and ϱ.  Therefore,  πϱ.

Title parallel and perpendicular planes
Canonical name ParallelAndPerpendicularPlanes
Date of creation 2013-04-19 15:18:51
Last modified on 2013-04-19 15:18:51
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 51M04
Related topic PlaneNormal
Related topic NormalOfPlane
Related topic ParallelismOfTwoPlanes