Question

In the adjoining diagram TA and TB are tangents, O is the centre. If ∠ PAT = 35° and ∠ PBT = 40°.
Calculate: 
(i) ∠ AQP,      (ii) ∠ BQP
(iii) ∠ AQB,    (iv) ∠ APB
(v) ∠ AOB,     (vi) ∠ ATB

Answer 1

(i) ∠ AQP = ∠ PAT = 35°     ....( Angles are in alternate segment)

(ii)  ∠ BQP = ∠ PBT = 40°    ....( Angles are in alternate segment)

(iii) ∠ AQB = ∠ AQP + ∠ BQP
∠ AQB = 35° + 40° = 75°

(iv) ∠ APB + ∠ AQB = 180°     ....(Opposite ∠s of a cyclic quadrilateral are supplementary)
∴ ∠ APB + 75° = 180°
∴ ∠ APB = 105°

(v) ∠ AOB = 2∠ AQB = 2(75°) = 150°   ....(Angle at the centre = 2 Angle at the circumference)

(vi) In quadrilateral AOBT:
∠ ATB = 360° - (∠OAT + ∠OBT + ∠AOB)
∠ ATB = 360° - (90° + 15° + 90°) = 30°   ....(∠OAT = ∠OBT = 90° radius, ⊥ tangent).