8-2. In square $A B C D$, a segment $C E$ is drawn such that the angles shown in the diagram are $7 \alpha$ and $8 \alpha$. Find the value of angle $\alpha$ in degrees. If necessary, round the answer to 0.01 or write the answer as a common fraction.

![](https://cdn.mathpix.com/cropped/2024_05_06_535cfcd84333341737d0g-15.jpg?height=534&width=602&top_left_y=1669&top_left_x=727)
Answer: $9^{\circ}$.

Solution. In triangle $D F E$, the angles are $7 \alpha, 8 \alpha$ and $45^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_535cfcd84333341737d0g-16.jpg?height=577&width=646&top_left_y=231&top_left_x=705)

Since the sum of the angles in triangle $D F E$ is $180^{\circ}$, we have $7 \alpha + 8 \alpha + 45^{\circ} = 180^{\circ}$, from which $\alpha = \frac{1}{15} \cdot 135^{\circ} = 9^{\circ}$.