Chapter 6 - Inequalities in Geometry

Property 1

If a > b, and c is equal to or more than d, then a+c > b+d

Property 2

If a > b and c > 0, then ac> bc and a/c > b/c

Property 3

If a > b and c < 0, then ac < bc and a/c < b/c.

Property 4

If a > b and b > c then a > c.

Property 5

If a = b+c and c > 0, then a > b.

Theorem: The exterior angle inequality theorem

The measure of an exterior angle is greater than the measure of either remote interior angle.

Statement

If p, then q

Inverse

If not p, then not q.

Contrapositive

If not q, then not p.

Converse

If q, then p.

Indirect proof

A proof technique that assumes the opposite of what you want to prove and shows that this assumption leads to a contradiction.

Theorem 6.2

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

Theorem 6.3

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

Corollary 1

The perpendicular segment from a point to a line is the shortest segment from that 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 6.4

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

Theorem: SAS inequality

If two sides of one triangle are congruent to 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.

Theorem: SSS inequality theorem

If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer 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.