10-6-1. In the figure, there are two circles with centers $A$ and $B$. Additionally, points $A, B$, and $C$ lie on the same line, and points $D, B$, and $E$ also lie on the same line. Find the degree measure of the angle marked with a “?”.

![](https://cdn.mathpix.com/cropped/2024_05_06_4212ab26029b1cb1f16eg-05.jpg?height=571&width=854&top_left_y=1231&top_left_x=607)
Answer: $24^{\circ}$.

Solution Variant 1. The inscribed angle $BFE$ is half the central angle $BAE$, so we will find the angle $BAE$. This angle can be found from the isosceles triangle $BAE$ ($AB=AE$ as radii of the circle). Angles $ABE$ and $DBC$ are equal as vertical angles. Note that triangle $DBC$ is isosceles with $BD$ and $BC$ equal as radii. Thus:

$$
\begin{gathered}
\angle ABE = \angle DBC = 180^{\circ} - 57^{\circ} - 57^{\circ} = 66^{\circ} \\
\angle BAE = 180^{\circ} - 66^{\circ} - 66^{\circ} = 48^{\circ} \\
\angle BFE = \frac{1}{2} \angle BAE = \frac{1}{2} \cdot 48^{\circ} = 24^{\circ}
\end{gathered}
$$