Question

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If $\angle DBC={70}^{\circ }$ and $\angle BAC={30}^{\circ }$, then $\angle BCD$= ___.

Answer 1

Given that $\angle DBC={70}^{\circ }$ and $\angle BAC={30}^{\circ }.$
Since the angles in the same segment are equal,
$\angle DBC=\angle DAC={70}^{\circ }.$
Now, $\angle DAC+\angle CAB=\angle DAB.$
$⟹\angle DAB={30}^{\circ }+{70}^{\circ }={100}^{\circ }$
Now since opposite angles are supplementary in a cyclic quadrilateral, we have
$\angle DAB+\angle BCD={180}^{\circ }.$
$⟹\angle BCD={180}^{\circ }-\angle DAB={180}^{\circ }-{100}^{\circ }={80}^{\circ }$