Question

ABCD is a parallelogram. The angle bisectors of $\angle$ A and $\angle$D meet at O. The measure of $\angle$AOD is:

• A. ${45}^{\circ }$
• B. ${90}^{\circ }$
• C. dependent on the angles A and D
• D. cannot be determined from given data
• E. none of these

Answer 1

The correct option is B ${90}^{\circ }$

Let $\angle A$ be $2x$
Let $\angle D$ be $2y$
Since $AB\parallel DC$ , sum of angles on same side of the transversal $AD$ will be ${180}^o$
$=>2x+2y={180}^o$
$x+y={90}^o$
After bisection of angles A and D, we get $\angle 1=x$ and $\angle 2=y$ as per the given figure.
Now in $△AOD$, we have
$x+y+\angle AOD={180}^o$
${90}^o+\angle AOD={180}^o$
$=>\angle AOD={90}^o$