[ Lines, rays, segments, and angles (miscellaneous).]

Inside the angle $A O B$, which is equal to $120^{\circ}$, rays $O C$ and $O D$ are drawn such that each of them is the bisector of one of the angles formed on the diagram. Find the measure of angle $AOC$, listing all possible variants.
Possible ray positions are shown in the figures.
![](https://cdn.mathpix.com/cropped/2024_05_06_b0b755a355dbd05a96cdg-20.jpg?height=682&width=1032&top_left_y=0&top_left_x=202)

1) If ray $O C$ is the bisector of angle $A O B$ (left figure), then $\angle A O C=60^{\circ}$ (regardless of whether ray $O D$ is the bisector of angle $A O C$ or angle $B O C$).
2) If ray $O D$ is the bisector of angle $A O B$, then $\angle A O C=30^{\circ}$ (if $O C$ is the bisector of angle $A O D$) or $\angle A O C=90^{\circ}$ (if $O C$ is the bisector of angle $\left.\angle B O D\right)$.
3) If ray $O C$ is the bisector of angle $A O D$, and ray $O D$ is the bisector of angle $B O C$, then $\angle A O C=60^{\circ}$.
4) Similarly, if ray $O D$ is the bisector of angle $\angle A O C$, and ray $O C$ is the bisector of angle $B O D$ (right figure), then $\angle A O C=80^{\circ}$.

## Answer

$30^{\circ}, 40^{\circ}, 60^{\circ}, 80^{\circ}$ or $90^{\circ}$.