In any triangle, the greater angle is subtended by the greater side. Let $ABC$ be a triangle having the angle $ABC$ greater than $BCA$. I say that side $AC$ is also greater than side $AB$.  For if not, $AC$ is certainly either equal to, or less than, $AB$. In fact, $AC$ is not equal to $AB$. For then angle $ABC$  would also have been equal to $ACB$ [Prop.~1.5]. But it is not. Thus, $AC$ is not equal to $AB$. Neither, indeed, is $AC$ less than $AB$. For then angle $ABC$ would also have been less than $ACB$ [Prop.~1.18]. But it is not. Thus, $AC$ is not less than $AB$. But it was  shown that ($AC$) is    not equal (to $AB$) either. Thus, $AC$  is greater than $AB$. Thus, in any triangle, the greater angle is subtended by the greater side. (Which is) the very thing it was required to show.