ABCD is a cyclic quadrilateral such that AB is the diameter of the circle circumscribing it and `/_ADC=155^@`, then what is the measure of `/_BAC`?

Options

A. `35^@`
B. `55^@`
C. `65^@`
D. `45^@`

Answer

To solve the problem, we need to find the measure of angle \( \angle BAC \) in the cyclic quadrilateral \( ABCD \) where \( AB \) is the diameter of the circle and \( \angle ADC = 155^\circ \).

### Step-by-Step Solution:

1. **Understand the Properties of Cyclic Quadrilaterals:**
- In a cyclic quadrilateral, opposite angles sum up to \( 180^\circ \).
- An angle subtended by a diameter at the circumference is a right angle (i.e., \( 90^\circ \)).

2. **Identify the Given Information:**
- We know that \( \angle ADC = 155^\circ \).

3. **Use the Property of Opposite Angles:**
- Since \( ABCD \) is a cyclic quadrilateral, we can use the property of opposite angles:
\[ \angle ADC + \angle ABC = 180^\circ \]
- Therefore, we can find \( \angle ABC \):
\[ \angle ABC = 180^\circ - \angle ADC = 180^\circ - 155^\circ = 25^\circ \]

4. **Determine \( \angle BAC \):**
- Since \( AB \) is the diameter, \( \angle ACB \) (which is the angle subtended by the diameter at point \( C \)) is \( 90^\circ \).
- Now, we can use the triangle sum property in triangle \( ABC \):
\[ \angle BAC + \angle ABC + \angle ACB = 180^\circ \]
- Substituting the known angles:
\[ \angle BAC + 25^\circ + 90^\circ = 180^\circ \]
- Simplifying this gives:
\[ \angle BAC + 115^\circ = 180^\circ \]
- Thus, we find \( \angle BAC \):
\[ \angle BAC = 180^\circ - 115^\circ = 65^\circ \]

### Conclusion:
The measure of \( \angle BAC \) is \( 65^\circ \).