Question

ABCD is a cyclic quadrilateral. If $\angle BCD={100}^{\circ }$ and $\angle ABD={70}^{\circ },$ then find $\angle ADB.$

• A. ${30}^{\circ }$
• B. ${40}^{\circ }$
• C. ${50}^{\circ }$
• D. ${60}^{\circ }$

Answer 1

The correct option is A ${30}^{\circ }$

Given that ABCD is a cyclic quadilateral such that$\angle BCD={100}^{\circ }$ and $\angle ABD={70}^{\circ }.$

Since opposite angles of a cyclic quadrilateral are supplementary,
$\angle DAB+\angle DCB={180}^{\circ }.$

$⟹\angle DAB={180}^{\circ }-\angle DCB$
$={180}^{\circ }-{100}^{\circ }={80}^{\circ }$

Consider $△ADB$, since sum of angles of a triangle is ${180}^{\circ },$ we have
$\angle ADB+\angle DAB+\angle ABD={180}^{\circ }$

$i.e.,\angle ADB+{80}^{\circ }+{70}^{\circ }={180}^{\circ }$

$⟹\angle ADB={180}^{\circ }-{150}^{\circ }={30}^{\circ }$