question_answer

ABCD is a parallelogram. The angle bisectors of \[\angle CPA={{45}^{o}}\] and \[\angle CBP\] meet at O. The measure of \[{{105}^{o}}\] is

A)  \[{{115}^{o}}\]

B)  \[{{135}^{o}}\]

C) dependent on the angles A and D

D) cannot be determined from given data

Correct Answer: B

Solution :

As shown in the figure the angle bisectors of \[\angle S={{75}^{o}}\] and \[\angle Q\]meet at O. Let \[{{50}^{o}}\] \[{{85}^{o}}\] \[{{120}^{o}}\] (\[{{360}^{o}}\] \[{{90}^{o}}\] and \[{{90}^{o}}\] and \[{{270}^{o}}\] are Adjacent angles) \[{{60}^{o}}\] \[{{300}^{o}}\] and  \[\therefore \frac{\angle D}{2}=\frac{180-x}{2}=90-\frac{x}{2}.\] \[\angle AOD=180-\left( \frac{x}{2}+90{}^\circ -\frac{x}{2} \right)\] \[\Rightarrow \angle AOD=180{}^\circ -(90{}^\circ )=90{}^\circ \]