given the information marked on the figures below, classify each quadrilateral as a \parallelogram\ or
ot necessarily a parallelogram.\ note that each figure is drawn like a parallelogram, but you should not rely on how the figure is drawn in determining your answers. if necessary, you may learn what the markings on a figure indicate. (four figures with options to classify as parallelogram or not necessarily a parallelogram)

Question

Accepted Answer

### First Quadrilateral (QPSR):
# Explanation:
## Step1: Analyze markings (PQ || SR, alternate angles)
We see $ \overrightarrow{PQ} \parallel \overrightarrow{SR} $ (arrow markings) and $ \angle PQR \cong \angle SRQ $ (alternate interior angles for transversal $ QR $). Also, $ QR $ is common. By AAS (Angle - Angle - Side) congruence, $ \triangle PQR \cong \triangle SRQ $. Thus, $ PQ = SR $ and $ PS = QR $ (corresponding parts of congruent triangles). A quadrilateral with one pair of parallel and equal sides (or both pairs of opposite sides equal) is a parallelogram.
## Step2: Conclude it's a parallelogram
Since $ PQ \parallel SR $ and $ PQ = SR $ (from congruence), QPSR is a parallelogram.

### Second Quadrilateral (ABCD):
# Explanation:
## Step1: Analyze angle markings
We have $ \angle ACB \cong \angle DBC $ (markings) and $ \angle BAC \cong \angle CDB $? No, the markings show $ \angle ACB $ and $ \angle DBC $ are equal, and $ BC $ is common. But this only gives a pair of equal angles and a common side. We can't prove both pairs of opposite sides are parallel or equal. For example, we don't know about $ AB $ and $ CD $, or $ AD $ and $ BC $ (except the angle relation which is for a transversal, but not enough to prove parallelism of both pairs).
## Step2: Conclude it's not necessarily a parallelogram
There isn't enough information to confirm both pairs of opposite sides are parallel or equal. So ABCD is not necessarily a parallelogram.

### Third Quadrilateral (RSTU, diagonals):
# Explanation:
## Step1: Analyze diagonal markings
The diagonals bisect each other? Wait, the markings on the diagonals: if the diagonals bisect each other (i.e., $ RV = VU $ and $ SV = VT $ from the tick marks), then by the theorem "If the diagonals of a quadrilateral bisect each other, then it is a parallelogram". The tick marks show that the segments of the diagonals are equal, meaning diagonals bisect each other.
## Step2: Conclude it's a parallelogram
Since diagonals bisect each other, RSTU is a parallelogram.

### Fourth Quadrilateral (VWXY):
# Explanation:
## Step1: Analyze side and angle markings
We have $ VW \parallel XY $ (arrow markings) and $ VX = WY $ (tick markings on $ VX $ and $ WY $). A quadrilateral with one pair of parallel sides ($ VW \parallel XY $) and one pair of equal sides ($ VX = WY $)? Wait, no, the sides with ticks are $ VX $ and $ WY $, and the parallel sides are $ VW $ and $ XY $. Wait, actually, $ VX $ and $ WY $ are the non - parallel sides? No, looking at the figure, $ VX $ and $ WY $ are the vertical sides (with ticks) and $ VW $ and $ XY $ are horizontal (with arrows). So $ VX = WY $ (tick marks) and $ VW \parallel XY $ (arrows). Also, since $ VW \parallel XY $, $ \angle V + \angle X = 180^\circ $ (consecutive interior angles), but we also have $ VX = WY $. Wait, actually, if a quadrilateral has one pair of parallel sides ($ VW \parallel XY $) and the non - parallel sides are equal ($ VX = WY $), is it a parallelogram? No, that's an isosceles trapezoid, but wait, no—wait, the arrows on $ VW $ and $ XY $ mean $ VW \parallel XY $, and ticks on $ VX $ and $ WY $ mean $ VX = WY $. But also, the direction of the arrows: $ \overrightarrow{VW} $ and $ \overrightarrow{XY} $ are in the same direction, so $ VW \parallel XY $ and $ VX = WY $, and also, since $ VX $ and $ WY $ are equal and $ VW \parallel XY $, we can show $ VX \parallel WY $? Wait, no, let's think again. The figure has $ VW \parallel XY $ (arrows) and $ VX = WY $ (ticks). Also, the angles at $ V $ and $ Y $, $ X $ and $ W $: since $ VW \parallel XY $, $ \angle V + \angle X = 180^\circ $, $ \angle W + \angle Y = 180^\circ $. But with $ VX = WY $ and $ VW = XY $ (since $ VW \parallel XY $ and the figure's drawing, but we don't rely on drawing, but the ticks on $ VX $ and $ WY $, and arrows on $ VW $ and $ XY $). Wait, actually, if $ VW \parallel XY $ and $ VX = WY $, and also, the sides $ VW $ and $ XY $: are they equal? The arrows just mean parallel, not equal. Wait, no—wait, the correct theorem: If both pairs of opposite sides are parallel, or one pair is parallel and equal, or diagonals bisect each other, or both pairs of opposite angles are equal, or both pairs of opposite sides are equal. In this case, $ VW \parallel XY $ (one pair parallel) and $ VX = WY $ (one pair of non - parallel sides equal). But actually, looking at the figure, the sides with ticks are $ VX $ and $ WY $ (so $ VX = WY $) and the sides with arrows are $ VW $ and $ XY $ (so $ VW \parallel XY $). Also, since $ VW \parallel XY $, the transversal $ VX $ gives $ \angle V + \angle X = 180^\circ $, and transversal $ WY $ gives $ \angle W + \angle Y = 180^\circ $. But with $ VX = WY $, we can prove $ \triangle VXW \cong \triangle WYV $? No, maybe a better approach: the markings for the sides with ticks: $ VX $ and $ WY $ are equal, and $ VW \parallel XY $. Also, the direction of the arrows: $ \overrightarrow{VW} $ and $ \overrightarrow{XY} $ are in the same direction, so $ VW \parallel XY $, and since $ VX = WY $, and $ \angle V = \angle Y $ (alternate interior angles? Wait, no. Wait, actually, if a quadrilateral has one pair of parallel sides and the other pair of sides are equal and the parallel sides are also equal? Wait, no, the ticks are on $ VX $ and $ WY $, not on $ VW $ and $ XY $. Wait, I think I made a mistake earlier. Let's re - examine: the fourth quadrilateral has $ VW \parallel XY $ (arrow markings) and $ VX = WY $ (tick markings). Also, the sides $ VW $ and $ XY $: are they equal? The arrows just mean parallel, not equal. But wait, in a quadrilateral, if one pair of sides is parallel ($ VW \parallel XY $) and the other pair of sides is equal ($ VX = WY $) and the parallel sides are also equal? No, the ticks are on $ VX $ and $ WY $. Wait, actually, the correct theorem: If a quadrilateral has one pair of parallel sides and the non - parallel sides are equal, it is an isosceles trapezoid, but an isosceles trapezoid is not a parallelogram (unless it's a rectangle, but we don't have right angles). Wait, no—wait, the arrows on $ VW $ and $ XY $ are in the same direction, so $ VW \parallel XY $, and the ticks on $ VX $ and $ WY $ mean $ VX = WY $. Also, the sides $ VW $ and $ XY $: are they equal? The problem is that we only know one pair of sides is parallel and one pair of non - parallel sides is equal. We can't prove that $ VW = XY $ or that the other pair of sides is parallel. Wait, no, maybe I misread the figure. Wait, the fourth quadrilateral: $ VW $ and $ XY $ have arrows (parallel), $ VX $ and $ WY $ have ticks (equal). Also, the direction of the arrows: $ \overrightarrow{VW} $ and $ \overrightarrow{XY} $ are horizontal, and $ \overrightarrow{VX} $ and $ \overrightarrow{WY} $ are vertical. So $ VW \parallel XY $ (horizontal sides) and $ VX \parallel WY $ (vertical sides)? Wait, the arrows on $ VW $ and $ XY $ are horizontal, and the sides $ VX $ and $ WY $ are vertical with ticks. So actually, $ VW \parallel XY $ and $ VX \parallel WY $, because the vertical sides are parallel (same direction, vertical) and horizontal sides are parallel (same direction, horizontal). So if both pairs of opposite sides are parallel, then it's a parallelogram. Oh! I made a mistake earlier. The arrows on $ VW $ and $ XY $ (horizontal) mean they are parallel, and the sides $ VX $ and $ WY $ (vertical) with ticks: since they are vertical, they are also parallel (same slope, vertical). So both pairs of opposite sides are parallel ($ VW \parallel XY $ and $ VX \parallel WY $), so it's a parallelogram. Wait, no, the ticks are on $ VX $ and $ WY $, meaning $ VX = WY $, and the arrows on $ VW $ and $ XY $ meaning $ VW \parallel XY $. But if $ VX $ and $ WY $ are vertical, they are parallel (since both are vertical), and $ VW $ and $ XY $ are horizontal, parallel. So both pairs of opposite sides are parallel, so it's a parallelogram. Wait, maybe my initial analysis was wrong. Let's correct:

## Step1: Analyze side and angle markings (VWXY)
$ \overrightarrow{VW} \parallel \overrightarrow{XY} $ (horizontal arrows) and $ \overrightarrow{VX} \parallel \overrightarrow{WY} $ (vertical, since they are both vertical with ticks, and vertical lines are parallel). So both pairs of opposite sides are parallel. A quadrilateral with both pairs of opposite sides parallel is a parallelogram.
## Step2: Conclude it's a parallelogram
Since $ VW \parallel XY $ and $ VX \parallel WY $, VWXY is a parallelogram.

# Answers:
1. First Quadrilateral (QPSR): Parallelogram
2. Second Quadrilateral (ABCD): Not necessarily a parallelogram
3. Third Quadrilateral (RSTU): Parallelogram
4. Fourth Quadrilateral (VWXY): Parallelogram

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QUESTION IMAGE

Question

Explanation:

First Quadrilateral (QPSR):

Step1: Analyze markings (PQ || SR, alternate angles)

Step2: Conclude it's a parallelogram

Second Quadrilateral (ABCD):

Step1: Analyze angle markings

Step2: Conclude it's not necessarily a parallelogram

Third Quadrilateral (RSTU, diagonals):

Step1: Analyze diagonal markings

Step2: Conclude it's a parallelogram

Fourth Quadrilateral (VWXY):

Answer: