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

Let $ABC$ be a triangle having the angle $ABC$ greater than the angle $BCA$;I say that the side $AC$ is also greater than the side $AB$.For, if not, $AC$ is either equal to $AB$ or less.Now $AC$ is not equal to $AB$;for then the angle $ABC$ would also have been equal to the angle $ACB$; [Prop. 1.5]but it is not;therefore $AC$ is not equal to $AB$.Neither is $AC$ less than $AB$,for then the angle $ABC$ would also have been less than the angle $ACB$; [Prop 1.18]but it is not;therefore $AC$ is not less than $AB$.And it was proved that it is not equal either.Therefore $AC$ is greater than $AB$.Therefore etc.Q.E.D.