Question

In figure (not to scale), the points M, R, N, S, Q are concyclic. Find $\angle PQR+\angle OPR+\angle NMS+\angle OSN$, if 'O' is centre of circle.

Answer 1

R.E.F image

To find $A+B+C+D$

In $△ORP,OR=OP=r\Rightarrow \angle ORP=\angle OPR=A$

Now, angle subtended by any are at

the center of circle is twise to that at

any point on circumference

By arc $NPS,\angle NOS=2\times \angle NMS=2D$

By arc $PS,\angle POS=2\times \angle PRS=2A$

By arc $RN,\angle RON=2\times \angle RSN=2B$

By arc $RNP,\angle ROP=2\times \angle RQP=2C$

now, by liner angle ${180}^{\circ },2C+2A={180}^{\circ }$ and

$2B+2D=180,$ adding both, we get

$2(A+B+C+D)={360}^{\circ }\Rightarrow A+B+C+D={180}^{\circ }$