Greater Angle of Triangle Subtended by Greater Side

Theorem

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

Proof

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 Triangles have 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.

$\blacksquare$

Historical Note

This is Proposition 19 of Book I of Euclid's The Elements.

This theorem is the converse of Proposition 18: Greater Side of Triangle Subtends Greater Angle.