If AB, BC and CD are equal chords of a circle with O as centre and AD diameter, then ∠AOB =

Question

Accepted Answer

To determine why Option 1 (60°) is the correct answer for the angle ∠AOB when AB, BC, and CD are equal chords of a circle with O as the center and AD as the diameter, we need to consider the properties of circles and the geometry involved.

### Explanation for Option 1 (60°):

1. **Equal Chords and Central Angles**: In a circle, equal chords subtend equal angles at the center. Since AB, BC, and CD are equal chords, the angles ∠AOB, ∠BOC, and ∠COD are equal.

2. **Sum of Angles in a Circle**: The total angle around point O, the center of the circle, is 360°. Since AD is the diameter, it divides the circle into two semicircles. Therefore, the sum of angles ∠AOB, ∠BOC, and ∠COD is 180° because they lie on one semicircle.

3. **Calculation of Each Angle**: Since ∠AOB = ∠BOC = ∠COD and their sum is 180°, we can set up the equation:
   $$
   3x = 180°
   $$
   Solving for x gives:
   $$
   x = 60°
   $$
   Therefore, ∠AOB = 60°.

### Explanation for Why Other Options are Incorrect:

- **Option 2 (90°)**: If ∠AOB were 90°, then the sum of ∠AOB, ∠BOC, and ∠COD would be 270°, which contradicts the fact that these angles must sum to 180° in a semicircle.

- **Option 3 (120°)**: If ∠AOB were 120°, then the sum of ∠AOB, ∠BOC, and ∠COD would be 360°, which is not possible since they are angles in a semicircle.

- **Option 4 (135°)**: If ∠AOB were 135°, then the sum of ∠AOB, ∠BOC, and ∠COD would be 405°, which exceeds the total angle around a point and is not possible in this context.

In conclusion, the only angle that satisfies the conditions of the problem, given that AB, BC, and CD are equal chords and AD is the diameter, is 60°. Therefore, Option 1 is the correct answer.