ABCD is a cyclic quadrilateral such that AB is a diameter of the circle circumscribing it and angle ADC=`146^@.angleBAC` is equal to:

Options

A. `18^@`
B. `24^@`
C. `72^@`
D. `56^@`

Answer

To solve the problem, we need to find the measure of angle BAC in the cyclic quadrilateral ABCD, where AB is the diameter of the circle circumscribing it, and angle ADC is given as 146 degrees.

### Step-by-Step Solution:

1. **Understanding the Properties of Cyclic Quadrilaterals**:
- In a cyclic quadrilateral, the sum of the opposite angles is equal to 180 degrees.
- Since AB is the diameter, angle ACB is a right angle (90 degrees) because the angle subtended by a diameter at the circumference is always 90 degrees.

2. **Using the Given Information**:
- We know that angle ADC = 146 degrees.

3. **Finding Angle ABC**:
- According to the property of cyclic quadrilaterals:
\[ \text{Angle ABC} + \text{Angle ADC} = 180^\circ \]
- Substituting the value of angle ADC:
\[ \text{Angle ABC} + 146^\circ = 180^\circ \]
- Solving for angle ABC:
\[ \text{Angle ABC} = 180^\circ - 146^\circ = 34^\circ \]

4. **Analyzing Triangle ABC**:
- In triangle ABC, we know that the sum of the angles is 180 degrees:
\[ \text{Angle BAC} + \text{Angle ACB} + \text{Angle ABC} = 180^\circ \]
- We already have:
- Angle ACB = 90 degrees
- Angle ABC = 34 degrees
- Substituting these values into the equation:
\[ \text{Angle BAC} + 90^\circ + 34^\circ = 180^\circ \]
- Simplifying the equation:
\[ \text{Angle BAC} + 124^\circ = 180^\circ \]
- Solving for angle BAC:
\[ \text{Angle BAC} = 180^\circ - 124^\circ = 56^\circ \]

5. **Conclusion**:
- Therefore, the measure of angle BAC is 56 degrees.

### Final Answer:
\[ \text{Angle BAC} = 56^\circ \]