Problem 7. In the given quadrilateral $A B C D$, no two sides are parallel. The angles between diagonal $A C$ and the sides of the quadrilateral are equal (in some order) to $16^{\circ}, 19^{\circ}, 55^{\circ}$ and $55^{\circ}$. What can the acute angle between the diagonals $A C$ and $B D$ be?
Solution. We will prove that the desired angle is equal to $87^{\circ}$. Let the intersection point of the diagonals be $E$. There are two fundamentally different configurations for the angles - either the equal angles share a vertex, or they have vertices at the ends of the diagonal and are on the same half-plane relative to that diagonal.

First, let $\Varangle B A C=\Varangle D A C=55^{\circ}, \Varangle B C A=19^{\circ}$ and $\Varangle D C A=16^{\circ}$.

The center $I$ of the inscribed circle in $\triangle A B D$ lies on the angle bisector $A C$. Since

$$
\Varangle B I D=90^{\circ}+\frac{1}{2} \Varangle B A D=145^{\circ}
$$

and $\Varangle B C D=16^{\circ}+19^{\circ}=35^{\circ}$, the quadrilateral $B C D I$ is cyclic. Therefore, $\Varangle D B I=\Varangle D C I=16^{\circ}$. From this, we find

![](https://cdn.mathpix.com/cropped/2024_06_03_2c6a040a9c526aedcd0ag-6.jpg?height=394&width=440&top_left_y=1225&top_left_x=1225)

$$
\begin{aligned}
& \Varangle A B D=2 \Varangle D B I=32^{\circ} \\
& \Varangle A E D=\Varangle A B D+\Varangle B A C=87^{\circ}
\end{aligned}
$$

The second case is $\Varangle B A C=\Varangle B C A=55^{\circ}, \Varangle D A C=19^{\circ}$ and $\Varangle D C A=16^{\circ}$. Then $\Varangle A D C=145^{\circ}, \Varangle A B C=70^{\circ}$. The center of the circumcircle of $\triangle A D C$ is the vertex of an isosceles triangle with base $A C$ and vertex angle equal to $2\left(180^{\circ}-\Varangle A D C\right)=70^{\circ}$. Therefore, point $B$ is the center of the circumcircle of $\triangle A D C$. From this, we find

$$
\begin{aligned}
& \Varangle A B D=2 \Varangle D C A=32^{\circ} \\
& \Varangle A E D=\Varangle A B D+\Varangle B A C=87^{\circ}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_06_03_2c6a040a9c526aedcd0ag-6.jpg?height=334&width=457&top_left_y=1832&top_left_x=1208)