In a cyclic quadrilateral ABCD, the side AB is extended to a point X. If `/_XBC=82^(@)` and `/_ADB=47^(@)` , then the value of `/_BDC` is

Options

A. `40^(@)`
B. `35^(@)`
C. `30^(@)`
D. `25^(@)`

Answer

To solve the problem, we need to find the value of angle \( \angle BDC \) in the cyclic quadrilateral \( ABCD \) given the angles \( \angle XBC = 82^\circ \) and \( \angle ADB = 47^\circ \).

### Step-by-Step Solution:

1. **Identify the angles**:
- We have \( \angle XBC = 82^\circ \).
- We have \( \angle ADB = 47^\circ \).

2. **Find \( \angle ABC \)**:
- Since \( AB \) is extended to point \( X \), the angles \( \angle CBX \) and \( \angle ABC \) are supplementary (they form a straight line).
- Therefore, we can write:
\[ \angle CBX + \angle ABC = 180^\circ \]
- Substituting the known value:
\[ 82^\circ + \angle ABC = 180^\circ \]
- Solving for \( \angle ABC \):
\[ \angle ABC = 180^\circ - 82^\circ = 98^\circ \]

3. **Use the property of cyclic quadrilaterals**:
- In a cyclic quadrilateral, the opposite angles are supplementary. Therefore:
\[ \angle ABC + \angle ADC = 180^\circ \]
- Substituting \( \angle ABC = 98^\circ \):
\[ 98^\circ + \angle ADC = 180^\circ \]
- Solving for \( \angle ADC \):
\[ \angle ADC = 180^\circ - 98^\circ = 82^\circ \]

4. **Find \( \angle BDC \)**:
- We know that \( \angle BDC \) can be found using:
\[ \angle BDC = \angle ADC - \angle ADB \]
- Substituting the known values:
\[ \angle BDC = 82^\circ - 47^\circ = 35^\circ \]

### Final Answer:
Thus, the value of \( \angle BDC \) is \( 35^\circ \).