Question

given that a quadrilateral pqrs is a parallelogram, pq and rs are opposite sides, pq = 7x + 9, rs = 3x + 17, and qr = 28 which of th more than one answer may be correct. following statements are correct? perimeter = 102 the parallelogram pqrs is a rhombus the diagonals are perpendicular x = 3 pq = 23 opposite angles are congruent none of these answer are correct

Explanation:

Step1: Use the property of parallelogram (opposite sides are equal)

In parallelogram \(PQRS\), \(PQ = RS\). So, \(7x + 9=3x + 17\).

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(7x-3x + 9=3x-3x + 17\), which gives \(4x+9 = 17\).
Subtract \(9\) from both sides: \(4x+9 - 9=17 - 9\), so \(4x=8\).
Divide both sides by \(4\): \(x=\frac{8}{4}=2\).

Step3: Calculate \(PQ\)

Substitute \(x = 2\) into \(PQ=7x + 9\). Then \(PQ=7\times2+9=14 + 9=23\).

Step4: Calculate the perimeter

The perimeter of a parallelogram \(P = 2(PQ + QR)\). First, find \(QR\). Since \(x = 2\), \(QR=3x + 17=3\times2+17=6 + 17=23\). Then \(P=2(23 + 23)=2\times46 = 92
eq102\).
A parallelogram is a rhombus if all sides are equal. Here \(PQ = QR = 23\), but we don't know about \(PS\) and \(QS\) from the given side - length equations (the equations only relate \(PQ\) and \(RS\), \(QR\) and \(PS\) would also need to be considered for a rhombus).
The diagonals of a parallelogram are perpendicular only if it is a rhombus (by the property of parallelograms: diagonals of a rhombus are perpendicular, but for a general parallelogram they are not).
Opposite angles of a parallelogram are congruent (by the property of parallelograms: in a parallelogram, opposite angles are equal).

Answer:

Opposite angles are congruent.