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, :