By definition, a circumscribed angle is an angle whose sides are tangents to a circle. Since $\angle ADB$ is a circumscribed angle, $\overrightarrow{DA}$ and $\overrightarrow{DB}$ are tangents to $\odot C$ at points $A$ and $B,$ respectively. By the Tangent to Circle Theorem, $\overline{CA}$ is perpendicular to $\overrightarrow{DA}$ and $\overline{CB}$ is perpendicular to $\overrightarrow{DB}.$

Notice that $ADBC$ is a quadrilateral and two of its angles are right angles. Recall that the sum of all of the angles in a quadrilateral is $360^{\circ }.$ Substituting the known angle measures and solving for $m\angle ADB$ will give the desired equation. This completes the proof.