ABCD is a cyclic quadrilateral and AD is a diameter. If `/_DAC = 55^@` then value of `/_ABC` is

Options

A. `55^@`
B. `35^@`
C. `145^@`
D. `125^@`

Answer

To solve the problem, we need to find the value of angle \( \angle ABC \) in the cyclic quadrilateral \( ABCD \) where \( AD \) is a diameter and \( \angle DAC = 55^\circ \).

### Step-by-Step Solution:

1. **Understanding the Properties of Cyclic Quadrilaterals**:
- In a cyclic quadrilateral, the opposite angles are supplementary. This means:
\[ \angle A + \angle C = 180^\circ \quad \text{and} \quad \angle B + \angle D = 180^\circ \]

2. **Using the Diameter Property**:
- Since \( AD \) is a diameter, angle \( \angle ACD \) (which is the angle subtended by the diameter at point \( C \)) is \( 90^\circ \). This is a property of circles: an angle inscribed in a semicircle is a right angle.

3. **Identifying the Angles**:
- We know that \( \angle DAC = 55^\circ \). Since \( \angle ACD = 90^\circ \), we can find \( \angle A \):
\[ \angle A = \angle DAC + \angle ACD = 55^\circ + 90^\circ = 145^\circ \]

4. **Finding \( \angle C \)**:
- Using the property of opposite angles in a cyclic quadrilateral:
\[ \angle A + \angle C = 180^\circ \]
Substituting \( \angle A = 145^\circ \):
\[ 145^\circ + \angle C = 180^\circ \]
\[ \angle C = 180^\circ - 145^\circ = 35^\circ \]

5. **Finding \( \angle ABC \)**:
- Now we can find \( \angle ABC \) using the property of angles in the cyclic quadrilateral:
\[ \angle B + \angle D = 180^\circ \]
- Since \( \angle D = \angle ACD = 90^\circ \):
\[ \angle B + 90^\circ = 180^\circ \]
\[ \angle B = 180^\circ - 90^\circ = 90^\circ \]

6. **Conclusion**:
- Therefore, the value of \( \angle ABC \) is \( 90^\circ \).

### Final Answer:
\[ \angle ABC = 90^\circ \]