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 determine the type of quadrilateral ABCD based on the angles given in the ratio 3:7:6:4. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Angles**:
The angles A, B, C, and D of quadrilateral ABCD are in the ratio 3:7:6:4. Let's denote the angles as:
- A = 3x
- B = 7x
- C = 6x
- D = 4x

2. **Using the Sum of Angles in a Quadrilateral**:
The sum of the interior angles of a quadrilateral is always 360 degrees. Therefore, we can write the equation:
\[ A + B + C + D = 360 \]
Substituting the expressions for A, B, C, and D:
\[ 3x + 7x + 6x + 4x = 360 \]

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

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

5. **Calculating Each Angle**:
Now we can find the measure of each angle:
- A = 3x = 3(18) = 54 degrees
- B = 7x = 7(18) = 126 degrees
- C = 6x = 6(18) = 108 degrees
- D = 4x = 4(18) = 72 degrees

6. **Analyzing the Angles**:
Now we have the angles:
- A = 54 degrees
- B = 126 degrees
- C = 108 degrees
- D = 72 degrees

7. **Determining the Type of Quadrilateral**:
To classify the quadrilateral, we check the properties of the angles:
- Since the sum of angles A and D (54 + 72 = 126) is equal to the angle B (126), we can conclude that sides AB and CD are parallel.
- Since only one pair of opposite angles is supplementary, ABCD is a trapezium.

### Conclusion:
Thus, quadrilateral ABCD is a **trapezium**.