If the angles P, Q, R and S of the quadrilateral PQRS are in the ratio 11: 19 : 21 : 9, then which of the following statements is true?

Options

A. PQRS is a parallelogram with `PQ || SR`.
B. PQRS is a trapezium with `PQ || SR`.
C. PQRS is a kite.
D. PQRS is a trapezium with `PS || QR`.

Answer

To solve the problem, we need to find the angles of the quadrilateral PQRS given that they are in the ratio of 11:19:21:9. Then, we will determine which statement about the quadrilateral is true.

### Step-by-Step Solution:

1. **Assign Variables Based on Ratios**:
Let the angles of the quadrilateral be:
- Angle P = 11x
- Angle Q = 19x
- Angle R = 21x
- Angle S = 9x

2. **Set Up the Equation**:
The sum of the angles in any quadrilateral is 360 degrees. Therefore, we can write the equation:
\[ 11x + 19x + 21x + 9x = 360 \]

3. **Combine Like Terms**:
Combine the terms on the left side:
\[ (11 + 19 + 21 + 9)x = 360 \]
This simplifies to:
\[ 60x = 360 \]

4. **Solve for x**:
Divide both sides by 60:
\[ x = \frac{360}{60} = 6 \]

5. **Calculate Each Angle**:
Now substitute x back into the expressions for each angle:
- Angle P = \(11x = 11 \times 6 = 66^\circ\)
- Angle Q = \(19x = 19 \times 6 = 114^\circ\)
- Angle R = \(21x = 21 \times 6 = 126^\circ\)
- Angle S = \(9x = 9 \times 6 = 54^\circ\)

6. **Check the Sum of Angles**:
Verify that the sum of the angles equals 360 degrees:
\[ 66 + 114 + 126 + 54 = 360 \]
This confirms our calculations are correct.

7. **Determine the Type of Quadrilateral**:
Since the angles are not all equal and one pair of opposite angles (P and Q) sum to 180 degrees, we can conclude that:
- PS is parallel to RQ (because angles P and Q are supplementary).
- PQ is not parallel to SR (as angles R and S are not supplementary).

Therefore, PQRS is a trapezium with PS parallel to RQ.

### Conclusion:
The correct statement is that PQRS is a trapezium with PS parallel to RQ.