Question

The angles $P,Q,R$ and $S$ of a quadrilateral are in the ratio $1:3:7:9$. Then $PQRS$ is a

• A. parallelogram
• B. trapezium with PQ || RS
• C. trapezium with QR||PS
• D. kite

Answer 1

The correct option is B

trapezium with PQ || RS

Step 1 - Finding the angles of a quadrilateral

We know that the Sum of all interior angles of quadrilaterals is $360°$

Let us assume the angles be $x,3x,7x,$ and $9x$

$$\begin{gathered} \Rightarrow x+3x+7x+9x=360 \\ \Rightarrow 20x=360 \\ \Rightarrow x=\frac{360}{20} \\ \Rightarrow x=18 \end{gathered}$$

Therefore the angles are

$$\begin{gathered} \angle P=x=18° \\ \angle Q=3x=3\times 18=54° \\ \angle R=7x=7\times 18=126° \\ \angle S=9x=9\times 18=162° \end{gathered}$$

The angles of the quadrilateral $PQRS$ are $18°,54°,126°$ and $162°$

Step 2 - Finding the sum of opposite angles

$$\begin{gathered} \angle P+\angle S=18°+162°=180° \\ \angle Q+\angle R=54°+126°=180° \end{gathered}$$

Therefore, the angle between the pair of parallel sides is supplementary.

So, $PQRS$ is a trapezium with $PQ\parallel RS$

Hence, Option B is correct