Question

State whether the following quadrilaterals are similar or not:

Answer 1

Similar Polygons:

Two polygons are said to be similar, if and only if,

$1.$ Their corresponding angles are equal.

$2.$ Their corresponding sides are in proportion (Same ratio).

Now lets check the given quadrilaterals are similar or not,

Since, Quadrilateral is one of the polygon,

In Quadrilateral $PQRS$ and $ABCD$,

In Quadrilateral $ABCD$,

$\Rightarrow \angle A=\angle B=\angle C=\angle D={90}^{\circ }$

But in Quadrilateral $PQRS$,

$\angle P\ne {90}^{\circ }$, $\angle Q\ne {90}^{\circ }$, $\angle R\ne {90}^{\circ }$ and $\angle S\ne {90}^{\circ }$

$\Rightarrow \angle A\ne \angle P,\angle B\ne \angle Q,\angle C\ne \angle R,\angle D\ne \angle S$

i.e., Corresponding angles are not equal. ---(1)

$\frac{AB}{PQ}=\frac{3cm}{1.5cm}=2$

$\frac{BC}{QR}=\frac{3cm}{1.5cm}=2$

$\frac{CD}{RS}=\frac{3cm}{1.5cm}=2$

$\frac{DA}{SP}=\frac{3cm}{1.5cm}=2$

$\Rightarrow \frac{AB}{PQ}=\frac{BC}{QR}=\frac{CD}{RS}=\frac{DA}{SP}$

i.e., Corresponding sides are in the same ratio.----(2)

From (1) and (2), Corresponding sides of the quadrilaterals are in the same ratio but their corresponding angles are not equal.

Hence, Quadrilateral $PQRS$ and $ABCD$ are not similar.