ABCD is a cyclic quadrilateral whose vertices are equidistant from the point O (centre of the circle). If `angleCOD = 120^@ ` and `angleBAC = 30^@`, then the measure of `angleBCD` is

Options

A. `180^@ `
B. `150 ^@`
C. `60^@`
D. `90^@`

Answer

To solve the problem, we need to find the measure of angle BCD in the cyclic quadrilateral ABCD, given that angle COD = 120° and angle BAC = 30°.

### Step-by-Step Solution:

1. **Understanding the Cyclic Quadrilateral**:
- A cyclic quadrilateral is a quadrilateral whose vertices lie on the circumference of a circle. The opposite angles of a cyclic quadrilateral sum up to 180°.

2. **Identify the Angles**:
- We are given:
- Angle COD = 120°
- Angle BAC = 30°
- We need to find angle BCD.

3. **Using the Central Angle Theorem**:
- The angle at the center (angle COD) is twice the angle at the circumference (angle BCD) that subtends the same arc.
- Therefore, if angle COD = 120°, then angle BCD = ½ * angle COD.
- So, angle BCD = ½ * 120° = 60°.

4. **Finding Angle DAB**:
- Now, we also know that angle BAC = 30°.
- Since angle DAB is subtended by the same arc as angle BCD, we can find angle DAB using the relationship between angles in a cyclic quadrilateral.
- Angle DAB = angle BAC + angle BCD = 30° + 60° = 90°.

5. **Using the Property of Cyclic Quadrilaterals**:
- The sum of opposite angles in a cyclic quadrilateral is 180°.
- Therefore, angle A + angle C = 180°.
- Here, angle A (DAB) = 90°, so angle C (BCD) = 180° - 90° = 90°.

6. **Conclusion**:
- Thus, the measure of angle BCD is 90°.

### Final Answer:
Angle BCD = 90°.

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