Question

In the following figure, ‘O’ is the centre of the circle. If ∠AOB = 40° and ∠BCD = 105°, find ∠OBD.

Answer 1

∠AOB = 40° is a central angle, so it subtends arc AB.

∠BCD = 105° is an inscribed angle subtending arc BD.

Step 1: Find arc AB

Central angle ∠AOB = 40°   ...[So arc AB = 40°]

Step 2: Find arc BD from ∠BCD

∠BCD = `1/2` × (arc BD)

105° = `1/2`  × are BD

arc BD = 210°

Step 3: Work out arc AD

arc AD = 360° − arc AB − arc BD

arc AD = 360° − 40° − 210°

arc AD = 110°

Step 4: Find ∠OBD

∠OBD subtends arc OD directly, and arc OD is:

arc OD = 360° − arc BO − arc BD

The angle at B between OB and BD subtends arc OD

Arc OD = 360° − (40° + 210°)

arc OD = 360° − 250° = 110°

∠OBD = `1/2`​ × 110° = 55°