1. **State the problem:** Find the measure of angle $\angle ADC$ in a quadrilateral $ABCD$ inscribed in circle $P$, given $\angle BAD = 110^\circ$ and $\angle ABC = 72^\circ$. 2. **Recall the property of cyclic quadrilaterals:** Opposite angles of a cyclic quadrilateral sum to $180^\circ$. 3. **Identify opposite angles:** $\angle BAD$ is opposite to $\angle BCD$, and $\angle ABC$ is opposite to $\angle ADC$. 4. **Calculate $\angle BCD$ using $\angle BAD$:** $$\angle BAD + \angle BCD = 180^\circ$$ $$110^\circ + \angle BCD = 180^\circ$$ $$\angle BCD = 180^\circ - 110^\circ = 70^\circ$$ 5. **Calculate $\angle ADC$ using $\angle ABC$:** $$\angle ABC + \angle ADC = 180^\circ$$ $$72^\circ + \angle ADC = 180^\circ$$ $$\angle ADC = 180^\circ - 72^\circ = 108^\circ$$ **Final answer:** $$\boxed{108^\circ}$$