In a triangle ABC, the median BD yields `angle BDC=45^@` and `angle ABD=angleBCD=x^@`. Find the value of x.

Options

A. `45^@`
B. `30^@`
C. `60^@`
D. none of these

Answer

To solve the problem, we need to find the value of \( x \) in triangle \( ABC \) given the conditions about the angles.

### Step-by-Step Solution:

1. **Identify the Angles**:
- We are given that \( \angle BDC = 45^\circ \).
- We also know that \( \angle ABD = \angle BCD = x^\circ \).

2. **Use the Triangle Angle Sum Property**:
- In triangle \( ABD \), the sum of the angles is \( 180^\circ \).
- Therefore, we can write the equation for the angles in triangle \( ABD \):
\[ \angle ABD + \angle ADB + \angle DAB = 180^\circ \]
- Substituting the known angles, we have:
\[ x + \angle ADB + \angle DAB = 180^\circ \]

3. **Express \( \angle ADB \) and \( \angle DAB \)**:
- Since \( BD \) is a median, \( D \) divides \( AC \) into two equal parts.
- This means that \( \angle ADB \) and \( \angle DAB \) can be expressed in terms of \( x \) and \( 45^\circ \).
- Specifically, \( \angle ADB = 45^\circ \) and \( \angle DAB = x \).

4. **Set Up the Equation**:
- Now substituting back into the angle sum equation:
\[ x + 45^\circ + x = 180^\circ \]
- Simplifying this gives:
\[ 2x + 45^\circ = 180^\circ \]

5. **Solve for \( x \)**:
- Rearranging the equation:
\[ 2x = 180^\circ - 45^\circ \]
\[ 2x = 135^\circ \]
- Dividing both sides by 2:
\[ x = \frac{135^\circ}{2} = 67.5^\circ \]

### Conclusion:
The value of \( x \) is \( 67.5^\circ \).