question_answer

The diagram shown is a quadrilateral ABCD inscribed in a circle. If AB = AD and CB = CD, then the measure of \[\angle A\] is

A)  \[\angle D\]                             

B)  \[\angle AOD\]

C)  \[{{45}^{o}}\]                   

D) indeterminate

Correct Answer: C

Solution :

First, we draw the diagonal AC. Now, in \[{{30}^{o}}\] and in \[\angle B={{90}^{o}},\angle C={{30}^{o}}\], we have \[\angle PXR=\angle UXS\]and \[{{180}^{o}}\](common) \[{{0}^{o}}\]  \[{{360}^{o}}\] \[{{90}^{o}}\] \[\angle DAB={{75}^{o}}\] coincides with \[\angle DBC={{60}^{o}}\] Since, ABCD is a cyclic quadrilateral in which sum of opposite angles is \[\angle CDB=....\] i.e.,  \[{{60}^{o}}\] \[{{75}^{o}}\]  Each angle = \[{{45}^{o}}\]\[{{135}^{o}}\] \[\sqrt{407}\] .