AB is a diameter of a circle and ABCD is a cyclic quadrilateral in that. If `angleACD=40^@`, then measurement of `angleBAD` is:

Options

A. `40^@`
B. `80^@`
C. `90^@`
D. `50^@`

Answer

To solve the problem, we need to find the measurement of angle \( \angle BAD \) given that \( \angle ACD = 40^\circ \) and \( AB \) is the diameter of the circle.

### Step-by-Step Solution:

1. **Understand the properties of the circle**: Since \( AB \) is the diameter, any angle inscribed in the semicircle will be a right angle. Thus, \( \angle ACB \) is \( 90^\circ \).

**Hint**: Remember that an angle inscribed in a semicircle is always a right angle.

2. **Identify the angles in the cyclic quadrilateral**: In cyclic quadrilaterals, the opposite angles are supplementary. This means that \( \angle ACD + \angle ABD = 180^\circ \).

**Hint**: Use the property of cyclic quadrilaterals where opposite angles sum to \( 180^\circ \).

3. **Calculate \( \angle ACB \)**: We already established that \( \angle ACB = 90^\circ \).

4. **Find \( \angle BCD \)**: Since \( \angle ACD = 40^\circ \), we can find \( \angle BCD \) using the triangle \( ACB \):
\[ \angle BCD = \angle ACB - \angle ACD = 90^\circ - 40^\circ = 50^\circ. \]

**Hint**: Remember that the angles in a triangle sum up to \( 180^\circ \).

5. **Use the property of cyclic quadrilaterals**: Now, we can use the cyclic quadrilateral property to find \( \angle BAD \):
\[ \angle BAD + \angle BCD = 180^\circ. \]
Substituting \( \angle BCD = 50^\circ \):
\[ \angle BAD + 50^\circ = 180^\circ. \]

6. **Solve for \( \angle BAD \)**:
\[ \angle BAD = 180^\circ - 50^\circ = 130^\circ. \]

**Hint**: Always isolate the variable to find its value.

7. **Final Calculation**: Thus, the measurement of \( \angle BAD \) is \( 130^\circ \).

### Conclusion:
The measurement of \( \angle BAD \) is \( 130^\circ \).