ABCD is a cyclic quadrilateral such that AB is a diameter of the circle circumscribing it and `angleADC=140^(@)," than "angleBAC ` is equal to

Options

A. `80^(@)`
B. `50^(@)`
C. `40^(@)`
D. `30^(@)`

Explanation

To solve the problem step by step, we will use properties of cyclic quadrilaterals and the fact that angles subtended by a diameter are right angles.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- ABCD is a cyclic quadrilateral.
- AB is the diameter of the circle.
- Angle ADC = 140°.

2. **Use the Property of Cyclic Quadrilaterals:**
- In a cyclic quadrilateral, the sum of the opposite angles is 180°.
- Therefore, we can write:
\[ \text{Angle ADC} + \text{Angle ABC} = 180° \]
- Substituting the known value:
\[ 140° + \text{Angle ABC} = 180° \]

3. **Solve for Angle ABC:**
- Rearranging the equation gives:
\[ \text{Angle ABC} = 180° - 140° = 40° \]

4. **Use the Diameter Property:**
- Since AB is the diameter, angle ACB is a right angle (90°) because an angle subtended by a diameter at the circumference is always 90°.

5. **Use the Triangle Angle Sum Property:**
- In triangle ACB, the sum of the angles is 180°:
\[ \text{Angle BAC} + \text{Angle ACB} + \text{Angle ABC} = 180° \]
- Substituting the known angles:
\[ \text{Angle BAC} + 90° + 40° = 180° \]

6. **Solve for Angle BAC:**
- Rearranging the equation gives:
\[ \text{Angle BAC} + 130° = 180° \]
- Therefore:
\[ \text{Angle BAC} = 180° - 130° = 50° \]

### Final Answer:
Angle BAC = 50°.

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