Question

In the given figure, the value of ${}^{\prime }{a}^{\prime }$ is:

• A. ${30}^{\circ }$
• B. ${40}^{\circ }$
• C. ${60}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is B ${40}^{\circ }$

Given- $\overline{AOB}$ is a diameter of a given circle. $ABCD$ is a cyclic quadrilateral. $AC$ is joined. Also, $\angle ADC={130}^{\circ }$.

Since, $\overline{AOB}$ is the diameter of the given circle, it subtends $\angle ACB$ to the circumference at $C$, i.e. $\angle ACB={90}^{\circ }$ ...[ since it is an angle in a semicircle].

Again,

$\angle ADC+\angle ABC={180}^{\circ }$ ...[ sum of opposite angles of a cyclic quadrilateral is ${180}^{\circ }$]

$\Rightarrow \angle ABC={180}^{\circ }-\angle ADC$

$\Rightarrow \angle ABC={180}^{\circ }-{130}^{\circ }$

$\Rightarrow \angle ABC={42}^{\circ }$.

In $△ABC,$

$\angle ABC+\angle ACB+\angle CAB={180}^{\circ }$

$\angle CAB={180}^{\circ }-(\angle ACB+\angle ABC)$

$={180}^{\circ }-({90}^{\circ }+{50}^{\circ })$

$={40}^{\circ }$

Hence, option $B$ is correct.