1. In a convex quadrilateral $A B C D$, $A B=B C=C D$, and $\angle A$ $=\angle B=\frac{1}{2} \angle C$. Find the measures of each interior angle.
(提示: 作 $\angle B$ 的平分线交 $A D$ 于点 $K$. 易得 $\triangle B C K \cong \triangle B A K, \triangle D C K \cong \triangle B C K$. 故 $\angle A=\angle B$ $\left.=80^{\circ}, \angle C=160^{\circ}, \angle D=40^{\circ}\right)$.

( Hint: Draw the angle bisector of $\angle B$ intersecting $A D$ at point $K$. It is easy to see that $\triangle B C K \cong \triangle B A K, \triangle D C K \cong \triangle B C K$. Therefore, $\angle A=\angle B$ $=80^{\circ}, \angle C=160^{\circ}, \angle D=40^{\circ}$ )