Question

Which of the following is true ($O$ is the centre of the circle)?

• A. $\angle AOB$ & $\angle COD$ are supplementary
• B. $\angle AOD$ & $\angle COD$ are supplementary
• C. $\angle BOC$ & $\angle COD$ are supplementary
• D. None of these

Answer 1

The correct option is A $\angle AOB$ & $\angle COD$ are supplementary

A circle with centre $O$, touches the sides $AB,BC,CD,DA,$ at $P,Q,R,S$respectively.
Join $OP,OQ,OR,OS$
Now two tangents drawn from an external C
point to a circle subtend equal angles at S
the centre.
$\therefore \left(i\right)AP=AS,\left(ii\right)BP=BQ,\left(iii\right)CR=CQ,\left(iv\right)DR=DS$
$\angle 1=\angle 2,\angle 3=\angle 4,\angle 5=\angle 6,\angle 7=\angle 8$
$\therefore \angle 1+\angle 2+\angle 3+\angle 4+\angle 5+\angle 6+\angle 7+\angle 8={360}^{\circ }$
(Sum of all angles subtended at a point is ${360}^{\circ }$)
$\Rightarrow 2(\angle 2+\angle 3+\angle 6+\angle 7)={360}^{\circ }$
$\Rightarrow (\angle 2+\angle 3)+(\angle 6+\angle 7)={180}^{\circ }$
$\Rightarrow \angle AOB+\angle COD={180}^{\circ }$
$\therefore \angle AOB,\angle COD$ are supplemntary angles.