The method of indirect proof will be used. First, the $\text{hypothesis}$ and the $\text{conclusion}$ of the theorem need to be identified. To start an indirect proof, the conclusion of the theorem needs to be assumed to be false. Next, any consequences of the assumption will be investigated. If a contradiction to the hypothesis is obtained, then the conclusion must be true. The assumption can be split into two parts. These two cases will be investigated separately.

$BC=AB$

If $BC$ equals $AB,$ then the following can be concluded about $△ABC.$

Claim	Explanation
$BC=AB$	Assumption
$\angle A\cong \angle C$	Isosceles Triangle Theorem
$m\angle A=m\angle C$	Congruent angles have the same measure

This contradicts the hypothesis, which states that the measure of $\angle A$ is greater than the measure of $\angle C.$

$BC<AB$

If $BC$ is less than $AB,$ the following can be concluded about $△ABC.$

Claim	Explanation
$BC<AB$	Assumption
$m\angle A<m\angle C$	Triangle Longer Side Theorem

Again, this contradicts the hypothesis, which states that the measure of $\angle A$ is greater than the measure of $\angle C.$

Conclusion

The assumption that $BC$ is less than or equal to $AB$ contradicts the hypothesis. Therefore, this assumption must be false. Consequently, the initial conclusion of the theorem is true.

It has been proven that if one angle of a triangle has a greater measure than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.