Adventitious angles problem: the lonely fractional derived angle

In the "classical" adventitious angle problem, for a given set of three angles $a$, $b$, and $c$ measured in integral degrees in an isosceles triangle, a fourth angle $\theta$ (the derived angle), also measured in integral degrees, is sought. We generalize the problem to find $\theta$ in fractional degrees. We show that the triplet $(a, b, c) = (45^\circ, 45^\circ, 15^\circ)$ is the only combination that leads to $\theta = 7\frac{1}{2}^\circ$ as the fractional derived angle.