In the figure below, the segments $A M, A K, B_{2} C_{2}$, and $B_{1} C_{1}$ are tangent to the circle with center $O$. If $\angle B_{1} O B_{2}=30^{\circ}$, determine the value of the angle $\angle B_{1} D B_{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_12f29789e10e6749ebb9g-24.jpg?height=529&width=1124&top_left_y=1224&top_left_x=358)
Solution

By the Exterior Angle Theorem, we have

$$
\begin{aligned}
\angle B_{1} O B_{2} & =\angle A B_{1} O-\angle A B_{2} O \\
& =\frac{\angle A B_{1} C_{1}-\angle A B_{2} C_{2}}{2} \\
& =\frac{\angle B_{1} D B_{2}}{2}
\end{aligned}
$$

Therefore, $\angle B_{1} D B_{2}=2 \cdot 30=60^{\circ}$.