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.

AC is tangent at A and OA is a radius. Therefore ∠BAC = 90°.

Given ∠ADB = 55°. Since A, D and C are collinear, in ΔABD: ∠ABD = 180° − 90° − 55° = 35°.

∠AOE = y is the central angle subtended by arc AE, while ∠ABE is the angle at the circumference on the same arc.

y = 2 × 35° = 70°.

In ΔAOC: ∠OAC = 90°, ∠AOC = 70°.

x = ∠ACO = 180° − 90° − 70° = 20°.

Answer: x = 20°, y = 70°.

The radius is perpendicular to the tangent at A. The central angle is twice the angle at the circumference standing on the same arc.