Question

choose all answers that describe the polygon pqrs if pq = 13, qr = 13, rs = 13, sp = 13, and m∠p = m∠q = m∠r = m∠s.
answer attempt 1 out of 3
parallelogram
quadrilateral
rectangle
rhombus
square
trapezoid

Explanation:

1. Quadrilateral: A polygon with four sides. Since \(PQ = QR=RS = SP = 13\), it has four sides, so it is a quadrilateral.
2. Parallelogram: A quadrilateral with both pairs of opposite sides equal and parallel, and opposite angles equal. Here, all sides are equal (so opposite sides are equal) and all angles are equal, so it satisfies the properties of a parallelogram.
3. Rectangle: A parallelogram with all angles equal to \(90^\circ\). Since all angles \(m\angle P=m\angle Q = m\angle R=m\angle S\), and the sum of interior angles of a quadrilateral is \(360^\circ\), each angle is \(\frac{360^\circ}{4}=90^\circ\), so it is a rectangle.
4. Rhombus: A parallelogram with all sides equal. Here, \(PQ = QR=RS = SP = 13\), so all sides are equal, hence it is a rhombus.
5. Square: A quadrilateral that is both a rectangle (all angles \(90^\circ\)) and a rhombus (all sides equal). Since it is a rectangle and a rhombus, it is a square.
6. Trapezoid: A quadrilateral with at least one pair of parallel sides. But in a parallelogram (which this is), both pairs of sides are parallel, but a trapezoid is usually defined (in some definitions) as having exactly one pair of parallel sides, so it does not fit the trapezoid definition (in the exclusive sense). So trapezoid is not a correct description.

Answer:

Quadrilateral, Parallelogram, Rectangle, Rhombus, Square