3 [ Quadrilateral: calculations, metric relations.]

Find the angles of the convex quadrilateral $A B C D$, in which $\angle B A C=30^{\circ}, \angle A C D=40^{\circ}, \angle A D B=50^{\circ}, \angle C B D$ $=60^{\circ}$ and $\angle A B C+\angle A D C=180^{\circ}$.
Answer: $\angle A B C=100^{\circ}, \angle A D C=80^{\circ}, \angle B A D=\angle B C D=90^{\circ}$. Since $\angle A B C+\angle A D C=180^{\circ}$, quadrilateral $A B C D$ is a cyclic quadrilateral. Therefore, $\angle A B C=\angle A B D+\angle C B D=\angle A C D+\angle C B D=40^{\circ}+60^{\circ}$ $=100^{\circ}$. Similarly, the other angles can be calculated.