ABCD is a cyclic trapezium with `AD||BC`, If `/_A = 105^@`, then other three angles are

Options

A. `/_B = 75^@, /_C = 75^@, /_D = 105^@`
B. `/_B = 105^@, /_C = 75^@, /_D = 75^@ `
C. `/_B = 75^@, /_C = 105^@, /_D = 75^@`
D. `/_B = 105^@, /_C = 105^@, /_D = 75^@`

Answer

To solve the problem, we need to find the angles B, C, and D of the cyclic trapezium ABCD, given that angle A is 105 degrees and that AD is parallel to BC.

### Step-by-Step Solution:

1. **Identify the given information:**
- We know that ABCD is a cyclic trapezium with AD || BC.
- Given angle A = 105 degrees.

2. **Use the property of cyclic trapeziums:**
- In a cyclic trapezium, the sum of the opposite angles is 180 degrees. Therefore, we can write:
\[ \angle A + \angle C = 180^\circ \]
- Substituting the value of angle A:
\[ 105^\circ + \angle C = 180^\circ \]

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

4. **Use the property of parallel lines:**
- Since AD is parallel to BC, by the properties of interior angles formed by a transversal, we have:
\[ \angle A + \angle B = 180^\circ \]
- Substituting the value of angle A:
\[ 105^\circ + \angle B = 180^\circ \]

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

6. **Find angle D using the cyclic trapezium property:**
- Again, using the property of opposite angles:
\[ \angle B + \angle D = 180^\circ \]
- Substituting the value of angle B:
\[ 75^\circ + \angle D = 180^\circ \]

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

### Summary of Angles:
- Angle A = 105 degrees
- Angle B = 75 degrees
- Angle C = 75 degrees
- Angle D = 105 degrees

### Final Answer:
- Angles B, C, and D are 75 degrees, 75 degrees, and 105 degrees respectively.