ABCD is a cyclic quadrilateral and O is the centre of the circle. If `/_COD = 140^@ `and `/_BAC = 40^@`. then the value of `/_BCD` is equal to

Options

A. `70^@`
B. `90^@`
C. `60^@`
D. `80^@`

Answer

To solve the problem, we need to find the value of angle BCD in the cyclic quadrilateral ABCD, given that angle COD = 140° and angle BAC = 40°.

### Step-by-Step Solution:

1. **Understanding Cyclic Quadrilateral:**
A cyclic quadrilateral is a four-sided figure where all vertices lie on the circumference of a circle. In a cyclic quadrilateral, the sum of the opposite angles is 180°.

2. **Identify Given Angles:**
We are given:
- Angle COD = 140°
- Angle BAC = 40°

3. **Finding Angle BCD:**
Since angle COD is the angle at the center of the circle, the angle subtended by the same arc (CD) at the circumference (angle BCD) is half of angle COD.
\[ \text{Angle BCD} = \frac{1}{2} \times \text{Angle COD} = \frac{1}{2} \times 140° = 70° \]

4. **Finding Angle ACD:**
Now, we need to find angle ACD. Since angle BAC = 40°, and we have found angle BCD = 70°, we can find angle ACD using the property of cyclic quadrilaterals:
\[ \text{Angle ACD} = 180° - \text{Angle BAC} - \text{Angle BCD} \]
\[ \text{Angle ACD} = 180° - 40° - 70° = 70° \]

5. **Finding Angle BCD Again:**
Now, we can find angle BCD again using the property of angles in a cyclic quadrilateral:
\[ \text{Angle BCD} + \text{Angle ACD} = 180° \]
Since we already calculated angle ACD = 70°, we can confirm:
\[ \text{Angle BCD} = 180° - \text{Angle ACD} = 180° - 70° = 110° \]

6. **Final Calculation:**
Thus, we find that:
\[ \text{Angle BCD} = 110° \]

### Conclusion:
The value of angle BCD is 110°.