Question

choose all answers that describe the quadrilateral qrst if qr = 42, rs = 85, st = 40, tq = 56, m∠q = 90°, m∠r = 81°, m∠s = 54°, and m∠t = 135°. answer attempt 1 out of 2 parallelogram quadrilateral rectangle rhombus square trapezoid

Explanation:

1. Quadrilateral: A quadrilateral is a four - sided polygon. Since \(QRST\) has four sides (\(QR = 42\), \(RS=85\), \(ST = 40\), \(TQ=56\)) and four angles (\(\angle Q = 90^{\circ}\), \(\angle R=81^{\circ}\), \(\angle S = 54^{\circ}\), \(\angle T=135^{\circ}\)), it is a quadrilateral.
2. Parallelogram: For a parallelogram, opposite sides are equal and opposite angles are equal. Here, \(QR = 42\), \(ST=40\) (not equal), \(TQ = 56\), \(RS = 85\) (not equal). Also, \(\angle Q=90^{\circ}\), \(\angle S = 54^{\circ}\) (not equal), \(\angle R = 81^{\circ}\), \(\angle T=135^{\circ}\) (not equal). So it is not a parallelogram.
3. Rectangle: A rectangle is a parallelogram with all angles \(90^{\circ}\). Since angles are not all \(90^{\circ}\) (e.g., \(\angle R = 81^{\circ}\)) and it is not a parallelogram, it is not a rectangle.
4. Rhombus: A rhombus is a parallelogram with all sides equal. Sides are not equal (\(QR = 42\), \(RS = 85\), etc.) and it is not a parallelogram, so not a rhombus.
5. Square: A square is a rhombus and a rectangle, so it is not a square.
6. Trapezoid: A trapezoid has at least one pair of parallel sides. Let's check the slopes (or use the angle - side relationship). In a trapezoid, if we consider the angles, for a pair of sides to be parallel, the consecutive angles should be supplementary (for a trapezoid with one pair of parallel sides). Let's check the sum of angles: \(\angle Q+\angle T=90^{\circ}+ 135^{\circ}=225^{\circ}

eq180^{\circ}\), \(\angle R+\angle S=81^{\circ}+54^{\circ}=135^{\circ}
eq180^{\circ}\), \(\angle Q+\angle R = 90^{\circ}+81^{\circ}=171^{\circ}
eq180^{\circ}\), \(\angle S+\angle T=54^{\circ}+135^{\circ}=189^{\circ}
eq180^{\circ}\). Wait, maybe we made a mistake. Wait, let's calculate the length of the diagonals or use the Pythagorean theorem on triangle \(QRS\) and \(TQR\). In triangle \(TQR\), since \(\angle Q = 90^{\circ}\), \(TR=\sqrt{QR^{2}+TQ^{2}}=\sqrt{42^{2}+56^{2}}=\sqrt{1764 + 3136}=\sqrt{4900}=70\). In triangle \(QRS\), \(QS=\sqrt{QR^{2}+RS^{2}-2\cdot QR\cdot RS\cdot\cos\angle R}=\sqrt{42^{2}+85^{2}-2\times42\times85\times\cos81^{\circ}}\). But maybe a better way: Wait, the sum of interior angles of a quadrilateral is \((4 - 2)\times180^{\circ}=360^{\circ}\). Let's check: \(90 + 81+54 + 135=360^{\circ}\), which is correct. Now, for a trapezoid, we can also check the sides. Let's see the ratio of sides. \(QR = 42\), \(TQ = 56\), \(ST = 40\), \(RS = 85\). Wait, maybe we misjudged. Wait, no, the key is that a quadrilateral is a four - sided figure, so it is a quadrilateral. Also, maybe we made a mistake in the trapezoid check. Wait, actually, the only one that must be true is quadrilateral, because it has four sides. The other special quadrilaterals have more conditions which are not met here.

Answer:

Quadrilateral