Which type of quadrilateral is formed when the angles A, B, C and D are in the ratio 2:4:5: 7?

Options

A. Rhombus
B. Square
C. trapezium
D. Rectangle

Answer

To determine the type of quadrilateral formed when the angles A, B, C, and D are in the ratio 2:4:5:7, we can follow these steps:

### Step-by-Step Solution:

1. **Assign Variables to Angles**:
- Let angle A = 2x
- Let angle B = 4x
- Let angle C = 5x
- Let angle D = 7x

2. **Use the Property of Quadrilaterals**:
- The sum of the angles in a quadrilateral is 360 degrees. Therefore, we can write the equation:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} + \text{Angle D} = 360^\circ \]
- Substituting the expressions for the angles:
\[ 2x + 4x + 5x + 7x = 360^\circ \]

3. **Combine Like Terms**:
- Combine the terms on the left side:
\[ 18x = 360^\circ \]

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

5. **Calculate Each Angle**:
- Now substitute the value of x back into the expressions for each angle:
- Angle A = \(2x = 2(20) = 40^\circ\)
- Angle B = \(4x = 4(20) = 80^\circ\)
- Angle C = \(5x = 5(20) = 100^\circ\)
- Angle D = \(7x = 7(20) = 140^\circ\)

6. **List the Angles**:
- The angles of the quadrilateral are:
- Angle A = 40°
- Angle B = 80°
- Angle C = 100°
- Angle D = 140°

7. **Determine the Type of Quadrilateral**:
- Check the properties of different quadrilaterals:
- **Rhombus**: Opposite angles must be equal. (Not satisfied)
- **Square**: All angles must be 90°. (Not satisfied)
- **Trapezium**: At least one pair of opposite sides is parallel, and the adjacent angles must be supplementary.
- Check pairs:
- Angle B (80°) + Angle C (100°) = 180° (supplementary)
- Angle A (40°) + Angle D (140°) = 180° (supplementary)
- Since both pairs of adjacent angles are supplementary, the quadrilateral is a trapezium.

### Final Answer:
The type of quadrilateral formed is a **trapezium**.