ABCD is a cyclic quadrilateral such that AB is a diameter of the circle circumscribing it and angle ADC = 150degree. Then angle BAC is equal to:

Options

A. a. 50degree
B. b. 30degree
C. c. 60degree
D. d. 40degree

Answer

To solve the problem, we will follow these steps:

1. **Understand the properties of cyclic quadrilaterals**: In a cyclic quadrilateral, the sum of the opposite angles is equal to 180 degrees.

2. **Identify the given information**: We know that angle ADC = 150 degrees.

3. **Use the property of cyclic quadrilaterals**: Since ABCD is a cyclic quadrilateral, we can write:
\[ \text{Angle ADC} + \text{Angle ABC} = 180^\circ \]
Substituting the known value:
\[ 150^\circ + \text{Angle ABC} = 180^\circ \]

4. **Solve for angle ABC**:
\[ \text{Angle ABC} = 180^\circ - 150^\circ = 30^\circ \]

5. **Identify the triangle formed**: We will now consider triangle ABC. Since AB is a diameter of the circle, angle ACB (the angle opposite to the diameter) is a right angle (90 degrees).

6. **Apply the triangle angle sum property**: The sum of angles in triangle ABC is:
\[ \text{Angle ABC} + \text{Angle ACB} + \text{Angle BAC} = 180^\circ \]
Substituting the known values:
\[ 30^\circ + 90^\circ + \text{Angle BAC} = 180^\circ \]

7. **Solve for angle BAC**:
\[ \text{Angle BAC} = 180^\circ - (30^\circ + 90^\circ) = 180^\circ - 120^\circ = 60^\circ \]

Thus, the value of angle BAC is \(60^\circ\).