In the figure (i) given below, PA and PB are tangents at the points A and B respectively of a circle with centre O. Q and R are points on the circle If ∠APB = 70°, find

(i) ∠AOB

(ii) ∠AQB

(iii) ∠ARB

AP and BP are tangents to the circle and OA and OB are radius of the circle.

∴ OA ⊥ AP and OB ⊥ BP.

∴ ∠OAP = ∠OBP = 90°

Sum of angles of a quadrilateral = 360°.

Hence, in quadrilateral OAPB,

⇒ ∠APB + ∠OAP + ∠OBP + ∠AOB = 360°
⇒ 70° + 90° + 90° + ∠AOB = 360°
⇒ 250° + ∠AOB = 360°
⇒ ∠AOB = 360° - 250°
⇒ ∠AOB = 110°.

Hence, the value of ∠AOB = 110°.

(ii) Arc AB subtends ∠AOB at centre and ∠AQB at remaining part of circle.

∴ ∠AOB = 2∠AQB (As angle at centre is double the angle subtended at remaining part of circle.)

⇒ 2∠AQB = 110°
⇒ ∠AQB = $\dfrac{110°}{2}$ = 55°.

Hence, the value of ∠AQB = 55°.

(iii) Reflex ∠AOB = 360° - ∠AOB = 360° - 110° = 250°.

Arc AB subtends Reflex ∠AOB at centre and ∠ARB at remaining part of circle.

∴ Reflex ∠AOB = 2∠ARB (As angle at centre is double the angle subtended at remaining part of circle.)

⇒ 2∠ARB = 250°
⇒ ∠ARB = $\dfrac{250°}{2}$ = 125°.

Hence, the value of ∠ARB = 125°.