10-3. Point $O$ is the center of the circle. What is the value of angle $x$ in degrees?

![](https://cdn.mathpix.com/cropped/2024_05_06_7c4dd4d2d6040292f233g-25.jpg?height=488&width=870&top_left_y=2269&top_left_x=593)
Answer: 9.

Solution. Since $O B=O C$, then $\angle B C O=32^{\circ}$. Therefore, to find angle $x$, it is sufficient to find angle $A C O: x=32^{\circ}-\angle A C O$.

![](https://cdn.mathpix.com/cropped/2024_05_06_7c4dd4d2d6040292f233g-26.jpg?height=497&width=897&top_left_y=437&top_left_x=585)

Since $O A=O C$, then $\angle A C O=\angle O A C=90^{\circ}-67^{\circ}=23^{\circ}$ (here we used the fact that triangle $A C D$ is a right triangle: angle $A C D$, which subtends the diameter, is a right angle).

Thus, $x=32^{\circ}-23^{\circ}=9^{\circ}$.