7.2. Angles $A O B, B O C$ and $C O D$ are equal to each other, while angle $A O D$ is three times smaller than each of them. All rays $\mathrm{OA}, \mathrm{OB}, \mathrm{OC}$, OD are distinct. Find the measure of angle $A O D$ (list all possible options).
# Solution.

Angles $A O B, B O C$ and $C O D$ follow each other in the same direction (since no rays coincide). Their sum can be less than $360^{\circ}$ (see Fig. 1) and more than $360^{\circ}$ (see Fig. 2).

Let the measure of angle AOD be denoted by x. Then each of the angles AOB, BOC, and COD is equal to 3x. In the first case, we get $3x + 3x + 3x + x = 360^{\circ}$, from which $x = 36^{\circ}$. In the second case, $3x + 3x + 3x - x = 360^{\circ}$, from which $x = 45^{\circ}$.

Fig. 1

![](https://cdn.mathpix.com/cropped/2024_05_06_16f9c3863cc94eacc671g-2.jpg?height=411&width=454&top_left_y=460&top_left_x=401)

Fig. 2

![](https://cdn.mathpix.com/cropped/2024_05_06_16f9c3863cc94eacc671g-2.jpg?height=380&width=512&top_left_y=481&top_left_x=1363)

Answer: $36^{\circ}, 45^{\circ}$

## Criteria:

- For one of the answers $36^{\circ}$ or $45^{\circ}$, accompanied by explanations, even in the form of a diagram - 3 points.
- Only one of the answers without explanations - 1 point. Both answers are written, but there are no explanations - 2 points.