Question

A quadrilateral $\text{ABCD}$ is inscribed in a circle. If $\angle \text{D}={55}^{\circ }$, then the $\angle \text{E}$ is

• A. ${55}^{\circ }$
• B. ${125}^{\circ }$
• C. ${65}^{\circ }$
• D. ${145}^{\circ }$

Answer 1

The correct option is A ${55}^{\circ }$

In the cyclic quadrilateral $\text{ABCD}$
Given that
$\angle \text{D}={55}^{\circ }$

Note: In cyclic quadrilateral, sum of opposite angles are supplementary.

Now, $\angle \text{B}$ and $\angle \text{D}$ are opposite to each other
$\Rightarrow \angle \text{B}+\angle \text{D}={180}^{\circ }$
$\angle \text{B}={180}^{\circ }-\angle \text{D}$
$={180}^{\circ }-{55}^{\circ }$
$={125}^{\circ }$

In the cyclic quadrilateral $\text{ABCE}$,
$\angle \text{B}$ and $\angle \text{E}$ are opposite to each other
$\Rightarrow \angle \text{E}+\angle \text{B}={180}^{\circ }$,
$\angle \text{E}={180}^{\circ }-\angle \text{B}$
$={180}^{\circ }-{125}^{\circ }$
$={55}^{\circ }$