ABCD is a cyclic quadrilateral such that AB is the diameter of the circle circumscribing it and `angleADC = 145^(@)` . What is the measure of `angle(BAC)` ?

Options

A. `50^(@)`
B. `35^(@)`
C. `55^(@)`
D. `40^(@)`

Answer

To solve the problem, we will use the properties of cyclic quadrilaterals and the fact that angles subtended by the same arc are equal.

### Step-by-Step Solution:

1. **Identify the cyclic quadrilateral**: We have a cyclic quadrilateral ABCD where AB is the diameter of the circle. Since AB is the diameter, angle ADB is a right angle (90 degrees) due to the inscribed angle theorem.

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

2. **Given angle**: We are given that angle ADC = 145 degrees.

3. **Calculate angle ADB**: Since AB is the diameter, angle ADB = 90 degrees.

4. **Use the property of cyclic quadrilaterals**: The sum of opposite angles in a cyclic quadrilateral is 180 degrees. Therefore, we can write:

\[ \text{Angle ADC} + \text{Angle ABC} = 180^\circ \]

Substituting the known value:

\[ 145^\circ + \text{Angle ABC} = 180^\circ \]

5. **Solve for angle ABC**:

\[ \text{Angle ABC} = 180^\circ - 145^\circ = 35^\circ \]

6. **Relate angle BAC to angle ABC**: Since angle BAC and angle ABC subtend the same arc AC, they are equal. Therefore:

\[ \text{Angle BAC} = \text{Angle ABC} = 35^\circ \]

### Final Answer:
The measure of angle BAC is **35 degrees**.