PQRS is a quadrilateral in which PQ = PS and RQ = RS. Which of the following statements is true about this quadrilateral ?

The correct option is: Its diagonals are perpendicular to each other.

Let us analyze the given quadrilateral $PQRS$ based on the given properties:

1. We are given that $PQ=PS$, which means point $P$ is equidistant from vertices $Q$ and $S$.
2. We are also given that $RQ=RS$, which means point $R$ is also equidistant from vertices $Q$ and $S$.

A quadrilateral with two pairs of equal adjacent sides is defined as a kite.

To determine the relationship between the diagonals $PR$ and $QS$, consider triangles $PQR$ and $PSR$:

• $PQ=PS$ (Given)
• $RQ=RS$ (Given)
• $PR=PR$ (Common side)

By the SSS (Side-Side-Side) congruence criterion, $△PQR\cong △PSR$.

Since the triangles are congruent, their corresponding angles are equal, so $\angle QPR=\angle SPR$.

Now let $O$ be the point of intersection of the diagonals $PR$ and $QS$. Consider triangles $POQ$ and $POS$:

• $PQ=PS$
• $\angle QPO=\angle SPO$
• $PO=PO$ (Common side)

By the SAS (Side-Angle-Side) congruence criterion, $△POQ\cong △POS$.

Therefore, $\angle POQ=\angle POS$.
Since $\angle POQ$ and $\angle POS$ form a linear pair:

$\angle POQ+\angle PO\mathrm{S<mo>=</mo><mn>180</mn><mo>°</mo>}$

$2\angle POQ=180°\Rightarrow \angle POQ=90°$

Thus, the diagonals $PR$ and $QS$ intersect at right angles, which means its diagonals are perpendicular to each other.