Relation between Angles and Sides of a Triangle

An important consequence of the corollary to the exterior angle theorem is this:

**Theorem 1:**

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

**Proof**:

Label the triangle as with , place point on side so , and draw segment :

Because , . But is isosceles, so , so (by substitution). Because is an exterior angle of , it is greater than the remote interior angle *A*: . So we have:

and

And by the transitive property for inequalities, it follows that . But is the angle opposite the longer side, , and is the angle opposite the shorter side, . Thus the angle opposite the longer side is greater than the angle opposite the shorter side.

The converse of this theorem is also true:

**Theorem 2:**

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.

**Proof:**

It is easier to prove the contrapositive of this statement:

If one side of a triangle, , is **not** longer than a second side, , then the angle opposite the first side, is **not** larger than the angle opposite the second side, .

So we assume in that side is **not** longer than side . Then either it is the same length, or it is shorter. If it is the same length, then we have an isosceles triangle, so . Therefore is **not** larger than .

If is **shorter** than side , then by Theorem 1, :