[ Inscribed Quadrilaterals [ Angles subtending equal arcs and equal chords $]$

In a convex quadrilateral $A B C D$, it is given that $\angle A B C=116^{\circ}, \angle A D C=64^{\circ}, \angle C A B=35^{\circ}$ and $\angle C A D=52^{\circ}$. Find the angle between the diagonals subtending side $A B$.

#
Let the diagonals of the given quadrilateral intersect at point $M$. Since

$$
\angle A B C+\angle A D C=116^{\circ}+64^{\circ}=180^{\circ},
$$

a circle can be circumscribed around the given quadrilateral. The desired angle $A M B$ is the exterior angle of triangle $A M D$. Therefore, it is equal to

$$
\begin{gathered}
\angle C A D+\angle A D B=52^{\circ}+\angle A D C-\angle B D C= \\
=52^{\circ}+64^{\circ}-35^{\circ}=81^{\circ}
\end{gathered}
$$

## Answer

$81^{\circ}$.