 Four points lying on one circle Auxiliary circle

In a convex quadrilateral $A B C D$, it is known that $\angle B C D=80^{\circ}, \angle A C B=50^{\circ}$ and $\angle A B D=30^{\circ}$. Find the angle $A D B$.
Quadrilateral $A B C D$ is convex, so ray $C A$ passes between the sides of angle $B C D$, hence,

$$
\angle A C D=\angle B C D-\angle A C B=80^{\circ}-50^{\circ}=30^{\circ}
$$

Points $B$ and $C$, lying on the same side of line $A D$, see segment $A D$ under the same angle, which means points $B, C, A$ and $D$ lie on the same circle. Inscribed angles $A D B$ and $A C B$ in this circle subtend the same arc, therefore,

$$
\angle A D B=\angle A C B=50^{\circ} .
$$

## Answer

$$
50^{\circ}
$$

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