jmc

We can determine the other two angles in the triangle with the unknown angle. Label its vertices $X$, $Y$, and $Z$.



We want to find $\angle XZY$. Since $\angle XYZ$ is an angle of a square, $\angle XYZ=90^{\circ }$. Also, $\angle YXZ$ is part of an angle of a square. By symmetry, the large angle at $X$ can be dissected into a sum of three angles, $$90^{\circ }=\angle YXZ+60^{\circ }+\angle YXZ=2\angle YXZ+60^{\circ }.$$ Therefore $\angle YXZ=15^{\circ }$. The mystery angle is the third angle of this triangle, so $$\angle XZY=180^{\circ }-90^{\circ }-15^{\circ }=\boxed{75^{\circ }}.$$