Question

In the given figure, O is the centre of the circle and AB is a diameter. If AC = BD and ∠AOC = 72°, find:

1. ∠ABC 
2. ∠BAD 
3. ∠ABD

Answer 1

Given:

O is the center of the circle.

AB is the diameter.

AC = BD.

∠AOC = 72°.

i. Find ∠ABC:

We know that the angle subtended by an arc at the centre of the circle is twice the angle subtended at any point on the remaining part of the circle.

Since AB is the diameter, ∠AOC subtends the arc AC.

Therefore, ∠ABC = `1/2 × ∠AOC = 1/2` × 72° = 36°.

ii. Find ∠BAD:

Given AC = BD (chords of the circle).

Equal chords subtend equal angles at any point on the major arc.

So, ∠BAD = ∠ABC = 36°.

iii. Find ∠ABD:

Since AB is the diameter, angle ∠ADB = 90° (angle in a semicircle).

In triangle ABD, sum of angles = 180°.

Therefore, ∠ABD + ∠BAD + ∠ADB = 180°

Substitute known values: ∠ABD + 36° + 90° = 180°

∠ABD = 180° − 126° = 54°.