Question

If PQ = RS and $\angle ORS={48}^{\circ }$ in the given figure, what are the values of $\angle OPQ$ and $\angle SOR$?

• A. ${84}^{\circ }$, ${48}^{\circ }$
• B. ${48}^{\circ }$, ${84}^{\circ }$
• C. ${38}^{\circ }$, ${68}^{\circ }$
• D. ${78}^{\circ }$, ${84}^{\circ }$

Answer 1

The correct option is B ${48}^{\circ }$, ${84}^{\circ }$

Consider the $△ORS.$
Note that OR = OS = radius of the circle. Since the angles opposite to equal sides are equal, we have $\angle ORS=\angle OSR={48}^{\circ }.$
But, using angle sum property
$\angle ORS+\angle RSO+\angle SOR={180}^{\circ }.$

i.e., ${48}^{\circ }+{48}^{\circ }+\angle SOR={180}^{\circ }$
$⟹\angle SOR={180}^{\circ }-{96}^{\circ }={84}^{\circ }$
Now, since equal chords subtend equal angles at the centre and since PQ=RS,
$\angle SOR=\angle POQ={84}^{\circ }.$
It can be noted that $△OPQ$ is also isosceles.
$⟹\angle OPQ=\angle OQP$
Thus, using angle sum property in $△OPQ,$
$\angle OPQ=\frac{{180}^{\circ }-{84}^{\circ }}{2}={48}^{\circ }.$