12.396 In a parallelogram, two sides $a$ and $b(a>b)$ and the acute angle $\alpha$ between the diagonals are given. Find the angles of the parallelogram.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-285.jpg?height=215&width=1001&top_left_y=632&top_left_x=233)

$$
\begin{aligned}
& A \\
& \Rightarrow 2 \cdot A \check{C} \cdot O K=a^{2}-b^{2} \Rightarrow S_{A B C D}=B K \quad A C= \\
& =O K \cdot \tan \alpha \cdot A C=\frac{a^{2}-b^{2}}{2} \cdot \tan \alpha
\end{aligned}
$$

On the other hand, $S_{A B C D}=a \cdot b \cdot \sin \angle A \Rightarrow \sin \angle A=\frac{a^{2}-b^{2}}{2 a b} \tan \alpha$.

Answer: $\arcsin \left(\frac{a^{2}-b^{2}}{2 a b} \tan \alpha\right), \pi-\arcsin \left(\frac{a^{2}-b^{2}}{2 a b} \tan \alpha\right)$.