Relation between Angles and Sides of a Triangle
An important consequence of the corollary to the exterior angle theorem is this:
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.
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 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:
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.
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, :