Question

State true or false:

In quadrilateral $PQRS$, $\angle P:\angle Q:\angle R:\angle S=3:4:6:7$. The Quadrilateral $PQRS$ is trapezium

• A. True
• B. False

Answer 1

The correct option is A True

Given in $\square$ PQRS,$\angle P:\angle Q:\angle R:\angle S=3:4:6:7$

Let $\angle P=3x,\angle Q=4x,\angle R=6x,\angle S=7x$

Sum of interior angles of a quadrilateral $={360}^o$

So $\angle P+\angle Q+\angle R+\angle S={360}^o$

$\Rightarrow 3x+4x+6x+7x={360}^o$

$\Rightarrow 20x={360}^o$

$\Rightarrow x=\frac{{360}^o}{20}$

$\Rightarrow x={18}^o$

$So\angle P=3x=3\times {18}^o={54}^o$

$\angle Q=4\times {18}^o={72}^o$

$\angle R=6\times {18}^o={108}^o$

$\angle S=7\times {18}^o={126}^o$

Now

$\angle P+\angle S={54}^o+{126}^o={180}^o\angle Q+\angle R={72}^o+{108}^o={180}^o$

In quadrilateral $PQRS$, $\angle Pand\angle S$ are supplementary as well as $\angle Qand\angle R$ are supplementary.

This is only possible when side PQ$\parallel$ SR ; $PS$ and $QR$ are transversals & the sum of interior corresponding angles on the same side of the transversals are supplementary.

So $PQ\parallel SR$.

Now $\angle P+\angle Q\ne {180}^{o}and\angle S+\angle R\ne {180}^o$

In quadrilateral PQRS, $\angle Pand\angle Q$ are not supplementary as well as $\angle Sand\angle R$ are not supplementary.

So $QR$ is not parallel to $SP$.

So one pair of opposite sides are parallel.

None of the opposite angles are equal.

None of the sides are given as equal.

The $\square$ PQRS can only be a Trapezium.