1. **Problem statement:** Given that $\angle ADC = \frac{1}{2} \angle AOC$ and $\angle ABC = 120^\circ$, find the value of $\angle AOC$. 2. **Understanding the problem:** The problem relates to angles formed by chords and arcs in a circle. The key formula here is that the measure of an inscribed angle is half the measure of its intercepted arc. 3. **Formula used:** For an inscribed angle $\angle ADC$, the measure is half the measure of the central angle $\angle AOC$ that subtends the same arc. $$\angle ADC = \frac{1}{2} \angle AOC$$ 4. **Given:** $\angle ABC = 120^\circ$. Since $\angle ABC$ is an inscribed angle subtending the same arc as $\angle ADC$, and $\angle ADC = \frac{1}{2} \angle AOC$, it follows that $\angle ADC = 120^\circ$. 5. **Calculate $\angle AOC$:** Using the formula: $$\angle ADC = \frac{1}{2} \angle AOC$$ Substitute $\angle ADC = 120^\circ$: $$120^\circ = \frac{1}{2} \angle AOC$$ Multiply both sides by 2: $$\angle AOC = 2 \times 120^\circ = 240^\circ$$ 6. **Final answer:** $\angle AOC = 240^\circ$. Note: The problem's answer states 120°, but based on the given relationship $\angle ADC = \frac{1}{2} \angle AOC$ and $\angle ABC = 120^\circ$, the central angle $\angle AOC$ should be $240^\circ$ if $\angle ADC = 120^\circ$. Possibly $\angle ABC$ and $\angle ADC$ are the same or related differently. However, following the formula strictly, $\angle AOC = 240^\circ$.