Question

In the given figure AC is a tangent to the circle with centre O.
If  ∠ADB = 55°, find x and y. Give reasons for your answers. 

Answer 1

In ΔABD, ∠BAD = 90°

∴ ∠ABD + ∠BAD + ∠ADB = 180°        ... [OA ⟂ AC because AC is a tangent, which justifies ∠BAD = 90° and ∠OAC = 90°.]

∴ ∠ABD + 90°  + 55°  =180° 

∴ ∠ABD = 35° 

Also ∠AOE = 2∠ABE = 2∠ABD

∴ ∠AOE = y = 2 × 35°

y = 70°

In  Δ AOC,

∠OAC + x + y = 180°     ...[OA ⟂ AC because AC is a tangent, which justifies ∠BAD = 90° and ∠OAC = 90°.]

∴ 90° + x + 70° = 180°

∴  90° + x + 70° = 180°

⇒ x = 20°