Question

In the given figure, ABCD is a parallelogram, OA, OB, OC and OD are the bisectors of $\angle A,\angle B,\angle C$ and $\angle D$ respectively.

The measure of $2(\alpha +\beta )–(x+y)$ is equal to

• A. ${90}^0$
• B. ${180}^0$
• C. ${270}^0$
• D. ${100}^0$

Answer 1

The correct option is B ${180}^0$

In a parallelogram, opposite angles are equal.
$\therefore \angle DAB=\angle DCB$ …..(i)
Since OA and OC are angle bisectors of $\angle A$ and $\angle C$ respectively.
$\therefore \angle BAO=\angle DAO$ and $\angle BCO=\angle DCO$
$\Rightarrow \angle BAO=\angle DAO=\angle BCO=\angle DCO$ [From (i)]
This implies that AOC is a straight line acting as a transversal for parallel lines AB and DC.
$\therefore \alpha +\beta =180°$ and $x+y=180°$
$\Rightarrow 2(\alpha +\beta )–(x+y)=2\times 180°–180°$
$=360°–180°$
$=180°$
Hence, the correct answer is option (b).