Unizor - Geometry3D - Lines and Planes - Angles between Lines and Planes

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Prior to discussing an angle between a line and a plane we have to define a very important concept of projection onto a plane.

Definition 1
If there is a plane γ and a point P outside this plane, a projection of point P onto plane γ is a point on plane γ which is a base of a perpendicular dropped from point P onto this plane.
If point P belongs to plane γ, its projection onto plane γ is this point itself.

Definition 2
If there is a plane γ and any curve c in the space, a projection of curve c onto plane γ is a set of projections of every individual point of this curve c onto plane γ.

Theorem 1
Projection of any straight line c onto any plane γ is a straight line on plane γ.

Note that a line and its projection onto any plane either intersect (if a line intersects a plane, since projection of a point of their intersection is this point itself) or are parallel (if the line is parallel to a plane).
Therefore, a line and its projection on any plane always belong to some plane, they are not skew lines.

Definition 3
An angle between a line and a plane is defined as an angle between this line and its projection onto this plane.

Theorem 2
Consider plane γ and line c that intersect this plane at point M (M=c∩γ).
Prove that an angle between line c and its projection onto plane γ (that is, an angle between line c and plane γ) is smaller than an angle between line c and any other line on plane γ that passes through point M and is not a projection of line c onto plane γ.