Question

In the given figure, $AB$ is a diameter of a circle with the centre $O$ and chord $ED$ is parallel to $AB$ and $\angle EAB={65}^o$. (i) $\angle EBA$ (ii) $\angle BED$ (iii) $\angle BCD$ are respectively:

• A. (i) ${155}^o$
  (ii) ${25}^o$
  (iii) ${25}^o$
• B. (i) ${65}^o$
  (ii) ${25}^o$
  (iii) ${155}^o$
• C. (i) ${155}^o$
  (ii) ${65}^o$
  (iii) ${25}^o$
• D. (i) ${25}^o$
  (ii) ${25}^o$
  (iii) ${155}^o$

Answer 1

The correct option is D (i) ${25}^o$

(ii) ${25}^o$
(iii) ${155}^o$

Since $AB$ is diameter,

$\angle AEB={90}^{\circ }$ ...[Angle formed in a semi circle].

In $△AEB$

$\angle ABE=180-\angle EAB-\angle AEB$ ...[Angle sum property]

$\angle ABE=180–65-90={25}^{\circ }$.

Given, $ED\parallel AB$.

$⟹\angle DEB=\angle EBA={25}^{\circ }$ ...[Alternate interior angles].

Since $EDCB$ is a cyclic quadrilateral opposite angles are supplementary

Then, $\angle DEB+\angle DCB={180}^{\circ }$

$⟹\angle DCB=180–25={155}^{\circ }$.

Hence, option $D$ is correct.