Question

What would be a situation in which the lengths of two sides of a right triangle are known, but the measure of an angle is needed.

Answer 1

You didn't clarify what the angle is needed for, but still, I'd say never.

If you know two sides of a right triangle you can always find the third using the pythagorean theorem.

Once all sides are known, you can find the three angles: one is surely right, since you have a right triangle. The other two can be found using the sine law.