Question

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

• A. ${30}^{\circ }$
• B. ${70}^{\circ }$
• C. ${80}^{\circ }$
• D. ${100}^{\circ }$

Answer 1

The correct option is C ${80}^{\circ }$

Given
$\angle DBC={70}^{\circ },\angle BAC={30}^{\circ }$
For chord CD,
$\angle CBD=\angle CAD\text{(Angles in the same segment)}\angle CAD={70}^{\circ }$
$\angle BAD=\angle BAC+\angle CAD={30}^{\circ }+{70}^{\circ }={100}^{\circ }$
$\angle BCD+\angle BAD={180}^{\circ }$ (Opposite angles of a cyclic quadrilateral)
$\angle BCD+{100}^{\circ }={180}^{\circ }$
$\angle BCD={80}^{\circ }$
Therefore, option c is correct.