1. **Problem statement:** Given that $\angle ADC = \frac{1}{2} \angle AOC$ and $\angle ABC = 120^\circ$, find the value of $\angle AOC$. 2. **Formula and rules:** In circle geometry, the angle at the circumference is half the angle at the center subtending the same arc. This means: $$\angle ADC = \frac{1}{2} \angle AOC$$ 3. **Given:** $\angle ADC = 120^\circ$ (since $\angle ABC = 120^\circ$ and $\angle ADC$ is the same angle on the circumference). 4. **Using the formula:** $$120^\circ = \frac{1}{2} \angle AOC$$ 5. **Solve for $\angle AOC$:** Multiply both sides by 2: $$\angle AOC = 2 \times 120^\circ = 240^\circ$$ 6. **Answer:** $$\boxed{240^\circ}$$ This means the central angle $\angle AOC$ is $240^\circ$ when the inscribed angle $\angle ADC$ is $120^\circ$.