Problem inside the circle

Since $GFDS$ is a cyclic quadrilateral, $\angle GSD + \angle GFD = 180^\circ \implies \angle SGD + \angle SDG = \angle GFD = 64^\circ$ . Also $\angle SGD = \dfrac {\widehat{SD}}2 = \dfrac {\frac {\widehat{FD}}2}2 = \dfrac {\angle DGF}2 = \dfrac {76^\circ}2 = 38^\circ$ . Therefore, $\angle SDG = 64^\circ - \angle SGD = 64^\circ - 38^\circ = \boxed{26}^\circ$ .

Let the center of the circle be $O$ . Then $\angle {DOF}=2\times \angle {DGF}=2\times \angle {SOD}$ (given). So $\angle {SOD}=76\degree$ , and hence $\angle {SGD}=38\degree$ . Since $SGFD$ is a cyclic quadrilateral, therefore $\angle {DSG}=180\degree-64\degree=116\degree$ . Hence $\angle {SDG}=180\degree-116\degree-38\degree=\boxed {26\degree}$