Section 5.4.

Shortest Distance Theorem- the perpendicular segment from a point to a line is the shortest segment from the point to the line.

Triangle Inequality Theorem- the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Corollary 5.1- The perpendicular segment from apoint to a plane is the shortest segment from the point to the plane.

Sec 5.5

Hinge Theorem (SAS Inequality)
-If two sides of one triangle are congruent to two sides of another triangle and the included angle of the 1st is larger than the included angle of the 2nd, then the 3rd side of the 1st is longer than the 3rd side of the 2nd.

Converse of Hinge Theorem (SSS Inequality)
-If two sides of one triangle are congruent to two sides of another triangle and the third side of the first is longer than the third side of the 2nd, then the included angle of the 1st is larger than the included angle of the 2nd.