Having proven the Scalene Triangle Inequality– that if in a scalene triangle ΔABC, AB>AC then m∠ACB> m∠ABC – proving the converse is very simple.
In scalene triangle ΔABC, m∠ACB> m∠ABC. Show that AB>AC.
When proving converse theorems, it is often useful to use proof by contradiction. Assume the converse theorem is not true, and see that in combination with the original theorem (which we know is true), it leads to a contradiction.
That means the assumption (that the converse is not true) is incorrect. That is what we will do here.
First assume that AC>AB. By the Scalene Triangle Inequality, we know that if AC>AB, then m∠ABC> m∠ACB. This contradicts what we were given – that m∠ACB> m∠ABC. So AC cannot be larger than AB.
Similarly, if we assume that AC=AB, then by the Base Angle Theorem, m∠ABC= m∠ACB, which contradicts what we were given – m∠ACB> m∠ABC. So AC cannot be equal to AB.
So if AC cannot be larger or equal to AB, it must be smaller.