Greater Angle of Triangle Subtended by Greater Side

Theorem
In any triangle, the greater angle is subtended by the greater side.

Proof

 * Euclid-I-19.png

Let $\triangle ABC$ be a triangle such that $\angle ABC$ is greater than $\angle BCA$.

Suppose $AC$ is not greater than $AB$.

If $AC$ were equal to $AB$, then by Isosceles Triangle has Two Equal Angles, $\angle ABC = \angle BCA$, but they're not so it isn't.

If $AC$ were less than $AB$, then by Greater Side of Triangle Subtends Greater Angle it would follow that $\angle ABC$ is less than $\angle BCA$, but it's not so it isn't.

So $AC$ must be greater than $AB$

Hence the result.