To find the angle measure of vertex C, we will use the Law of Cosines to first find angle A:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c = 9 (side opposite angle C), a = 12 (side opposite angle A), b = 15 (side opposite angle B), and C = angle C.
Plugging in the values:
9^2 = 12^2 + 15^2 - 2(12)(15) * cos(C)
81 = 144 + 225 - 360 * cos(C)
81 = 369 - 360 * cos(C)
360 * cos(C) = 288
cos(C) = 288 / 360
cos(C) = 0.8
Now, we will use the inverse tangent function to find the angle measure of C:
C = arctan(0.8)
C ≈ 37 degrees
Therefore, the approximate angle measure of vertex C is 37 degrees. Choose the answer option 37 degrees.