Question

parallelogram
not necessarily a parallelogram
parallelogram
not necessarily a parallelogram
parallelogram
not necessarily a parallelogram
parallelogram
not necessarily a parallelogram

Explanation:

Step1: Recall parallelogram properties

A parallelogram has opposite sides parallel and equal, and diagonals bisect each other.

Step2: Analyze first figure (PQRS)

Since \(PQ\parallel SR\) and \(PS = QR\), by the property of a parallelogram (one pair of opposite sides is parallel and equal), it is a parallelogram.

Step3: Analyze second figure (KLMN)

Since diagonals \(KM\) and \(LN\) bisect each other (\(KO = OM\) and \(LO=ON\)), by the property of a parallelogram (diagonals bisect each other), it is a parallelogram.

Step4: Analyze third figure (ABCD)

Since \(\angle ACB=\angle CBD\) (alternate - interior angles) and \(BC\) is common, \(\triangle ABC\cong\triangle CDB\) (by ASA if we consider \(AB\parallel CD\) assumption, but more simply, using the property that if one pair of opposite sides is parallel and the included angles are equal for the transversal, and we can also show \(AB = CD\) and \(AD = BC\) through congruence. So it is a parallelogram.

Step5: Analyze fourth figure (GHIJ)

Only \(GH\parallel IJ\) is given. There is no information about the other pair of sides (\(GI\) and \(HJ\)) being parallel or about the sides being equal. So it is not necessarily a parallelogram.

Answer:

First figure: Parallelogram
Second figure: Parallelogram
Third figure: Parallelogram
Fourth figure: Not necessarily a parallelogram