Geometric Inequalities

Indirect Proofs:
Assume temporarily that the conclusion is false and reason logically until a contradiction of the hypothesis or another fact is reached.

Theorem:
If one side of a triangle is longer than a second side, then the angle opposite the longer side is larger than the opposite angle of the second side.

Theorem:
If one angle of a triangle is larger than a second angle, then the side opposite the larger angle is longer than the opposite side of the second angle.

Corollary 1:
The perpendicular segment from a point to a line is the shortest segment from the point to the line.

Corollary 2:
The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.

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

Theorem: The SAS Inequality Theorem:
If two sides of a triangle is congruent with two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second triangle then, the third side of the first triangle is longer than the third side of the second triangle.

Theorem: The SSS Inequality Theorem:
If two sides of a triangle is congruent with two sides of another triangle, but the third side of the first triangle is larger than the third side of the second triangle then, the included angle of the first triangle is larger than the included angle of the second triangle.