Question

\(ABCD\) is a cyclic quadrilateral whose diagonals intersect at \(E\). If \(\angle DBC=70^\circ\), \(\angle BAC=30^\circ\) and \(AB=BC\), then \(\angle ECD=\)

• A. \(30^\circ\)
• B. \(50^\circ\)
• C. \(60^\circ\)
• D. \(70^\circ\)

Hint:

In cyclic quadrilateral problems, always check for equal angles subtended by the same chord.

Answer 1

The Correct Option is B

Since: \[ AB=BC \] triangle \(ABC\) is isosceles. Given: \[ \angle BAC=30^\circ \] So: \[ \angle ACB=30^\circ \] Thus: \[ \angle ABC=180^\circ-(30^\circ+30^\circ)=120^\circ \] Given: \[ \angle DBC=70^\circ \] Hence: \[ \angle ABD=\angle ABC-\angle DBC=120^\circ-70^\circ=50^\circ \] In a cyclic quadrilateral, angles subtended by the same chord are equal. Both: \[ \angle ABD \] and \[ \angle ACD \] stand on chord \(AD\). So: \[ \angle ACD=50^\circ \] Since \(E\) lies on diagonal \(AC\), \[ \angle ECD=\angle ACD=50^\circ \] Thus, \[ \boxed{50^\circ} \]