1. **State the problem:** In circle H, we are given two angles: $m\angle DHG = (11x - 36)^\circ$ and $m\angle GHF = (x + 12)^\circ$. We need to find the measure of arc $DG$. 2. **Recall the rule:** In a circle, the measure of the arc intercepted by two radii forming a central angle is equal to the measure of that central angle. 3. **Find $x$:** Since $\angle DHG$ and $\angle GHF$ are adjacent central angles that together form the central angle $\angle DHF$, which is a straight angle (diameter), their measures add up to $180^\circ$: $$ (11x - 36) + (x + 12) = 180 $$ Simplify: $$ 11x - 36 + x + 12 = 180 $$ $$ 12x - 24 = 180 $$ Add 24 to both sides: $$ 12x - \cancel{24} + 24 = 180 + 24 $$ $$ 12x = 204 $$ Divide both sides by 12: $$ \frac{\cancel{12}x}{\cancel{12}} = \frac{204}{12} $$ $$ x = 17 $$ 4. **Find $m\angle DHG$:** $$ m\angle DHG = 11x - 36 = 11(17) - 36 = 187 - 36 = 151^\circ $$ 5. **Find $mDG$:** Since $\angle DHG$ is a central angle intercepting arc $DG$, the measure of arc $DG$ is equal to $m\angle DHG$: $$ mDG = 151^\circ $$ **Final answer:** $$ \boxed{151^\circ} $$