ABCD is a cyclic trapezium whose sides AD and BC are parallel to each other. If `/_ABC = 75^@` then the measure of `/_BCD` is:

Options

A. `75^@`
B. `95^@`
C. `45^@`
D. `105^@`

Answer

To solve the problem, we need to find the measure of angle BCD in the cyclic trapezium ABCD, where AD and BC are parallel, and angle ABC is given as 75 degrees.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- We have a cyclic trapezium ABCD.
- The sides AD and BC are parallel.
- Angle ABC = 75 degrees.

2. **Use the Property of Cyclic Quadrilaterals:**
- In a cyclic quadrilateral, the sum of the opposite angles is 180 degrees.
- Therefore, we can write:
\[ \angle ABC + \angle ADC = 180^\circ \]

3. **Substitute the Known Angle:**
- We know that angle ABC = 75 degrees. Substitute this into the equation:
\[ 75^\circ + \angle ADC = 180^\circ \]

4. **Solve for Angle ADC:**
- Rearranging the equation gives:
\[ \angle ADC = 180^\circ - 75^\circ = 105^\circ \]

5. **Use the Property of Parallel Lines:**
- Since AD is parallel to BC, the angles ADC and BCD are supplementary:
\[ \angle ADC + \angle BCD = 180^\circ \]

6. **Substitute Angle ADC:**
- We found that angle ADC = 105 degrees. Substitute this into the equation:
\[ 105^\circ + \angle BCD = 180^\circ \]

7. **Solve for Angle BCD:**
- Rearranging gives:
\[ \angle BCD = 180^\circ - 105^\circ = 75^\circ \]

### Final Answer:
The measure of angle BCD is **75 degrees**.

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