Question

$ABCD$ is a cyclic quadrilateral and $PQ$ is a tangent to the circle at $C$. If $BD$ is a diameter and $\angle DCQ={40}^{\circ }$ and $\angle ABD={60}^{\circ }$; find the measure of the following angles
$\angle DBC$
$\angle BCP$
$\angle ADB$.

• A. ${30}^{\circ }$
  ${45}^{\circ }$
  ${70}^{\circ }$
• B. ${20}^{\circ }$
  ${40}^{\circ }$
  ${90}^{\circ }$
• C. ${40}^{\circ }$
  ${50}^{\circ }$
  ${30}^{\circ }$
• D. None of these
  

Answer 1

Answer: (C)

${40}^{\circ }$

${50}^{\circ }$
${30}^{\circ }$
Since $BD$ is a diameter of the circle.
$\therefore \angle BAD={90}^{\circ }$

and also $\angle BCD={90}^{\circ }$

$\angle DBC=\angle DCQ={40}^{\circ }$

[$\angle$s in the alternate segments]
$\angle BCP+\angle BCD+\angle DCQ={180}^{\circ }$

$\Rightarrow \angle BCP+{90}^{\circ }+{40}^{\circ }={180}^{\circ }$

$\angle BCP={50}^{\circ }$

Similarly from $\Delta BAD,{60}^{\circ }+\angle BAD+\angle ADB={180}^{\circ }$

$\Rightarrow {60}^{\circ }+{90}^{\circ }+\angle ADB={180}^{\circ }$

$\angle ADB={30}^{\circ }$.