If angles A, B, C and D of the quadrilateral ABCD, taken in order are in the ratio 3 : 7 : 6 : 4, then ABCD is a

Options

A. rhombus
B. parallelogram
C. trapezium
D. kite

Explanation

To solve the problem, we need to find the angles of quadrilateral ABCD given that they are in the ratio 3:7:6:4. We will then determine the nature of the quadrilateral based on these angles.

### Step-by-Step Solution:

1. **Set Up the Ratios**:
Let the angles A, B, C, and D be represented as:
- Angle A = 3x
- Angle B = 7x
- Angle C = 6x
- Angle D = 4x

2. **Sum of Angles in a Quadrilateral**:
The sum of the angles in any quadrilateral is 360 degrees. Therefore, we can write the equation:
\[ 3x + 7x + 6x + 4x = 360 \]

3. **Combine Like Terms**:
Combine the terms on the left side:
\[ (3 + 7 + 6 + 4)x = 360 \]
\[ 20x = 360 \]

4. **Solve for x**:
Divide both sides by 20 to find x:
\[ x = \frac{360}{20} = 18 \]

5. **Calculate Each Angle**:
Now substitute x back into the expressions for the angles:
- Angle A = \(3x = 3 \times 18 = 54^\circ\)
- Angle B = \(7x = 7 \times 18 = 126^\circ\)
- Angle C = \(6x = 6 \times 18 = 108^\circ\)
- Angle D = \(4x = 4 \times 18 = 72^\circ\)

6. **Check the Sum of Angles**:
Verify that the sum of the angles equals 360 degrees:
\[ 54 + 126 + 108 + 72 = 360 \]

7. **Determine the Type of Quadrilateral**:
To determine the type of quadrilateral, we can check the properties of the angles:
- Angle A + Angle B = \(54 + 126 = 180^\circ\)
- Angle C + Angle D = \(108 + 72 = 180^\circ\)

Since the sum of each pair of opposite angles is 180 degrees, ABCD is a **cyclic quadrilateral**.

### Conclusion:
Quadrilateral ABCD is a cyclic quadrilateral.