In the figure (ii) given below, O is the centre of the circle. PS and PT are tangents and ∠SPT = 84°. Calculate the sizes of the angles TOS and TQS.

Since PS and PT are tangents on the circle. So,

∠OSP = ∠OTP = 90°.

Since, sum of angles in a quadrilateral = 360

∠TOS + ∠OTP + ∠SPT + ∠OSP = 360°
∠TOS + 90° + 84° + 90° = 360°
∠TOS + 264° = 360°
∠TOS = 360° - 264°
∠TOS = 96°.

Reflex ∠TOS = 360° - ∠TOS = 360° - 96° = 264°.

Arc ST subtends reflex ∠TOS at the centre and ∠TQS at the remaining part of the circle.

∴ Reflex ∠TOS = 2∠TQS

∠TQS = $\dfrac{1}{2}$ x Reflex ∠TOS = $\dfrac{1}{2}$ x 264° = 132°.

Hence, the value of ∠TOS = 96° and ∠TQS = 132°.