Question 6

We will use properties of cyclic quadrilaterals and parallelograms.

Step 1 — Identify the shape from diagonals

We are given that diagonals bisect each other. This is a key property of parallelograms. So, PQRS must be a parallelogram. In a parallelogram, opposite angles are equal. Let's say $\angle P = \angle R$. Also, $\angle Q = \angle S$.

Step 2 — Use the cyclic property

We are also given that PQRS is cyclic. In a cyclic quadrilateral, opposite angles sum to $\mathbf{180^\circ}$. So, $\angle P + \angle R = \mathbf{180^\circ}$. From Step 1, we know $\angle P = \angle R$. Let's substitute this into the equation. $\angle P + \angle P = 180^\circ$ $2 \angle P = 180^\circ$ $\angle P = \frac{180^\circ}{2}$ $\angle P = 90^\circ$ Since $\angle P = \angle R$, then $\angle R = \mathbf{90^\circ}$. In a parallelogram, consecutive angles sum to $\mathbf{180^\circ}$. So, $\angle P + \angle Q = \mathbf{180^\circ}$. $90^\circ + \angle Q = 180^\circ$ $\angle Q = 180^\circ - 90^\circ$ $\angle Q = 90^\circ$ Since $\angle Q = \angle S$, then $\angle S = \mathbf{90^\circ}$. All angles of the quadrilateral are $\mathbf{90^\circ}$.

$\boxed{\text{All angles of PQRS are } 90^\circ}$

Step 3 — Conclude the quadrilateral type

PQRS is a parallelogram. All its interior angles are $\mathbf{90^\circ}$. A parallelogram with all angles $\mathbf{90^\circ}$ is a rectangle. Therefore, PQRS is a rectangle.

Answer

The quadrilateral PQRS is a rectangle.