Question

According to the figure shown, which of the following is correct?

• A. $\alpha =\beta ={55}^{\circ }$
• B. $\alpha +\beta ={110}^{\circ }$
• C. Both $A\&C$
• D. $\theta ={110}^{\circ }$

Answer 1

Answer: (C)

Refer figure:

Construct: Join points $B$ and $D$ at $A$ such that $A$ lies on the circumference.

Join $B\text{and}D$

Thus, $\angle A=\frac{1}{2}\times \angle O$ ( angle subtended by a chord at the center is twice the angle at the circumference)

thus, $\angle A=\frac{140}{2}=70°$

Also, $ABCD$ is a cyclic quadrilateral. Hence, $\angle A+\Theta =180°$ (sum of opposite angles of a cyclic quadrilateral is 180 degrees)

$\Theta =180-70=110°$

In $△OBD$,

$OB=OD$ (radius of circle)

$\angle OBD=\angle ODB=\frac{1}{2}\times (180-140)$ (angles opposite to equal sides are equal)

Hence, $\angle OBD=\angle ODB=20°$

In $△DCB$

$\theta +\angle CBD+\angle CDB=180°$

$\theta +(\alpha -\angle OBD)+(\beta -\angle ODB)=180°$

$110°+\alpha -20°+\beta -20°=180°$

Thus, $\alpha +\beta =110°$

Summarizing,

$\theta =110°$

$\alpha +\beta =110°$

Thus, both options A&C are correct.

Correct choice : D