In the given figure, O is the centre of the circle. If ∠OAC = 55°, then ∠OBD = ?

1. 55°

2. 35°

3. 45°

4. 70°

In ΔOAC,

OA = OC [Radii of same circle]

Since, two sides are equal it is isosceles triangle and opposite sides are also equal:

∠OCA = ∠OAC = 55°

In ΔOBD,

OB = OD [Radii of same circle]

Since, two sides are equal it is isosceles triangle and opposite sides are also equal:

∠OBD = ∠ODB

∠AOC = ∠BOD [vertically opposite angles]

In ΔOAC,

By angle sum property of triangle,

∠AOC + ∠OAC + ∠OCA = 180°

∠AOC + 55° + 55° = 180°

∠AOC + 110° = 180°

∠AOC = 180° - 110°

∠AOC = 70°

∠AOC = ∠BOD = 70°

In ΔOBD,

By angle sum property of triangle,

∠BOD + ∠OBD + ∠ODB = 180°

70° + 2∠OBD = 180°

2∠OBD = 180° - 70°

2∠OBD = 110°

∠OBD = $\dfrac{110^{\circ}}{2}$

∠OBD = 55°.

Hence, option 1 is the correct option.