ABCD is a cyclic quadrilateral such that AB is diameter of the circle circumscirbing it and angle `ADC=126^@. angle BAC` is equal to

Options

A. `72^@`
B. `36^@`
C. `18^@`
D. `24^@`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the properties of the cyclic quadrilateral
Given that ABCD is a cyclic quadrilateral with AB as the diameter, we can use the properties of cyclic quadrilaterals. One important property is that the opposite angles of a cyclic quadrilateral sum up to 180 degrees.

### Step 2: Identify the given angle
We are given that angle ADC = 126 degrees.

### Step 3: Use the property of cyclic quadrilaterals
Since ABCD is a cyclic quadrilateral, we can write:
\[ \angle ABC + \angle ADC = 180^\circ \]
Substituting the value of angle ADC:
\[ \angle ABC + 126^\circ = 180^\circ \]

### Step 4: Solve for angle ABC
To find angle ABC, we rearrange the equation:
\[ \angle ABC = 180^\circ - 126^\circ = 54^\circ \]

### Step 5: Analyze triangle ABC
Now, we will consider triangle ABC. We know that:
- Angle ABC = 54 degrees
- Angle ACB (the angle opposite to AB, which is the diameter) is a right angle (90 degrees) because angle subtended by a diameter in a circle is a right angle.

### Step 6: Use the triangle angle sum property
In triangle ABC, the sum of angles is 180 degrees:
\[ \angle ABC + \angle ACB + \angle BAC = 180^\circ \]
Substituting the known values:
\[ 54^\circ + 90^\circ + \angle BAC = 180^\circ \]

### Step 7: Solve for angle BAC
Rearranging gives us:
\[ \angle BAC = 180^\circ - 54^\circ - 90^\circ = 36^\circ \]

### Conclusion
Thus, the measure of angle BAC is:
\[ \angle BAC = 36^\circ \]