What is the measure of the vertex angle of an isosceles triangle if one of its base angles measures 75 and its congruent sides each measure 15 cm?

The length of the congruent sides is irrelevant.

If one of the base angles is 75 deg, the two base angles sum to 150 degrees. Since the sum of all three angles is 180 degrees, the third angle, the vertex angle, must be 180 - 150 = 30 degrees.