Final answer:
The measure of angle ZC in right triangle ABC where sin A = sin B is 90 degrees given that angles A and B would both be 45 degrees and angle C is the right angle.
Step-by-step explanation:
To find the measure of angle ZC in right triangle ABC where sin A = sin B, first note that since ABC is a right triangle, angles A and B are complementary, meaning they add up to 90 degrees (since the third angle C is the right angle). The sine function is positive in the first and second quadrants, and since A and B are acute angles in a right triangle, they must both be in the first quadrant where sine values are positive. If sin A = sin B, and both angles are complementary, it's only possible when A = B = 45 degrees. This is because the sine of an angle in a right triangle is equal to the opposite side over the hypotenuse and, for the sides opposite to angles A and B to be equal (implying equal sine values), the triangle must also be isosceles.
Therefore, if A = B = 45 degrees, angle C being the right angle is 90 degrees. Knowing that the sum of angles in a triangle is 180 degrees, we can say:
Measure of angle ZC = 180 degrees - angle A - angle B = 180 - 45 - 45 = 90 degrees.
Thus, the measure of angle ZC is 90 degrees which is not among the provided options. There might have been a typo in the question or in the options provided.