ABCD is a cyclic trapezium with AB || DC and AB = diameter of the circle. If `angleCAB = 30^@`, then `angleADC` is

Options

A. `60^@`
B. `120^@`
C. `150^@`
D. `30^@`

Answer

To solve the problem, we will use the properties of cyclic trapeziums and the angles formed by the diameter of the circle.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have a cyclic trapezium ABCD where AB is parallel to DC and AB is the diameter of the circle.
- Since AB is the diameter, angle ACB is a right angle (90 degrees) according to the inscribed angle theorem.

2. **Identifying Given Angles**:
- We are given that angle CAB = 30 degrees.

3. **Finding Angle ABC**:
- In triangle ABC, we know that the sum of angles in a triangle is 180 degrees.
- Therefore, angle ABC can be calculated as:
\[ \text{angle ABC} = 180^\circ - \text{angle CAB} - \text{angle ACB} \]
\[ \text{angle ABC} = 180^\circ - 30^\circ - 90^\circ = 60^\circ \]

4. **Using the Properties of Cyclic Quadrilaterals**:
- In a cyclic quadrilateral, the opposite angles are supplementary. Thus, angle ABC + angle ADC = 180 degrees.
- We already found that angle ABC = 60 degrees.

5. **Calculating Angle ADC**:
- Using the supplementary angle property:
\[ \text{angle ADC} = 180^\circ - \text{angle ABC} \]
\[ \text{angle ADC} = 180^\circ - 60^\circ = 120^\circ \]

### Final Answer:
Thus, angle ADC is 120 degrees.