Question

for each figure, determine if it is a parallelogram. first figure (q, s, r, p with diagonals intersecting at t): options are parallelogram, not necessarily a parallelogram. second figure (w, y, x, v): options are parallelogram, not necessarily a parallelogram. third figure (b, d, c, a): options are parallelogram, not necessarily a parallelogram. fourth figure (s, u, t, k): options are parallelogram, not necessarily a parallelogram.

Explanation:

Step1: Analyze Top - Left Figure (PQRS)

In a quadrilateral, if the diagonals bisect each other, then the quadrilateral is a parallelogram. In the figure PQRS, the diagonals PR and QS intersect at T. We can see that \( PT = TR \) (marked with single ticks) and \( QT=TS \) (marked with double ticks). So the diagonals bisect each other. By the theorem "If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram", PQRS is a parallelogram. So we select "Parallelogram" for the top - left figure.

Step2: Analyze Top - Right Figure (VWXY)

In a quadrilateral, if a pair of opposite sides are both parallel and congruent, then the quadrilateral is a parallelogram. In the figure VWXY, we see that \( VW\parallel XY \) (both have upward arrows, indicating parallel lines) and \( VW = XY \) (both have single ticks, indicating congruent segments). So, by the theorem "If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram", VWXY is a parallelogram. So we select "Parallelogram" for the top - right figure.

Step3: Analyze Bottom - Left Figure (ABCD)

We know that \( AB\parallel CD \) (both have upward arrows). Also, we can prove that \( \triangle ABC\cong\triangle CDA \) (using ASA or AAS, since \( \angle BAC=\angle DCA \) (alternate interior angles as \( AB\parallel CD \)), \( AC = AC \) (common side), and \( \angle BCA=\angle DAC \) (alternate interior angles as \( AB\parallel CD \))). So \( AB = CD \) and \( AB\parallel CD \). By the theorem "If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram", ABCD is a parallelogram. So we select "Parallelogram" for the bottom - left figure.

Step4: Analyze Bottom - Right Figure (SKTU)

We know that \( \angle S=\angle K \) and \( \angle U=\angle T \), but this does not necessarily imply that the opposite sides are parallel or congruent. Just having two pairs of equal angles does not guarantee that the quadrilateral is a parallelogram (for example, an isosceles trapezoid has base angles equal but is not a parallelogram in the general sense of having both pairs of opposite sides parallel, unless it is a rectangle). So we cannot be sure that SKTU is a parallelogram. So we select "Not necessarily a parallelogram" for the bottom - right figure.

Step5: Summarize the Selections

• Top - Left: Parallelogram
• Top - Right: Parallelogram
• Bottom - Left: Parallelogram
• Bottom - Right: Not necessarily a parallelogram

Answer:

Top - Left: Parallelogram
Top - Right: Parallelogram
Bottom - Left: Parallelogram
Bottom - Right: Not necessarily a parallelogram