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

We know that angle between the radius and tangent at the point of contact is right angle.

∴ ∠A = 90°.

Also in △OBE, OB = OE = radius of the circle.

∴ ∠B = ∠OEB …..(i)

In △ABD,

⇒ ∠A + ∠B + ∠ADB = 180°
⇒ 90° + ∠B + 55° = 180°
⇒ ∠B + 145° = 180°
⇒ ∠B = 180° - 145° = 35°.

∴ ∠OEB = 35°.

From figure,

∠DEC = ∠OEB = 35° (∵ vertically opposite angles are equal.)

∠EDC + ∠ADE = 180° (∵ both form a linear pair)

∠EDC + 55° = 180°
∠EDC = 180° - 55°
∠EDC = 125°.

In △EDC,

⇒ ∠DEC + ∠EDC + ∠DCE = 180°
⇒ 35° + 125° + x° = 180°
⇒ x° + 160° = 180°
⇒ x° = 180° - 160° = 20°.

In △AOC,

⇒ ∠AOC + ∠OAC + ∠ACO = 180°
⇒ y° + 90° + x° = 180°
⇒ y° + 90° + 20° = 180°
⇒ y° + 110° = 180°
⇒ y° = 180° - 110° = 70°.

Hence, the value of x = 20 and y = 70.