Two results follow from Theorems 1 and 2:

Corollary 1:

The shortest distance from a point to a line is the length of the perpendicular segment from that point to that line.

Proof:

Let  be a point not on the line, let  be the perpendicular segment from  to the line, and suppose point  is some other point on the line.  Draw segment .  Then  is a right triangle with .  Since the angles of a triangle add to 180^o, the other two angles in this triangle must be less than 90^o.  Therefore, , so the opposite sides have the same inequality:  .  That is, any other segment from  to the line is longer than .

Corollary 2:

The perpendicular segment from a point to a plane is the shortest distance from that point to the plane.

Proof:

Let  be a point not in the plane, let  be the perpendicular segment from  to the plane, and suppose point  is some other point in the plane.  Then  is shorter than  because it is the shortest segment to line , by Corollary 1.