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. **Understand the properties of cyclic quadrilaterals**: In a cyclic quadrilateral, the opposite angles are supplementary. This means that the sum of the angles at opposite vertices equals 180°.

2. **Identify the angles**: We are given:
- Angle COD = 120°
- Angle BAC = 30°

3. **Use the inscribed angle theorem**: The angle subtended by an arc at the center of the circle (angle COD) is twice the angle subtended at any point on the circumference (angle BAC). Therefore, angle BCD will be equal to angle COD.

4. **Calculate angle BCD**: Since angle COD = 120°, angle BCD will also be equal to 120°.

5. **Find angle ACD**: Since angle BAC is given as 30°, we can find angle ACD using the property of cyclic quadrilaterals:
- Angle ACD = 180° - angle BAC = 180° - 30° = 150°.

6. **Use the triangle properties**: In triangle ODC, we know:
- Angle ODC + Angle OCD + Angle COD = 180°
- Let angle ODC = x and angle OCD = x (since they are equal).
- Therefore, we have: x + x + 120° = 180°.
- This simplifies to 2x = 60° → x = 30°.

7. **Determine angle BCD**: Now we know:
- Angle ACD = 30° (from step 6).
- Angle BCA = 60° (from the cyclic property).
- Thus, angle BCD = angle ACD + angle BCA = 30° + 60° = 90°.

### Final Answer:
The measure of angle BCD is **90°**.