using the given information below, which test for parallelograms proves that hijk is a parallelogram? image of quadrilateral hijk with markings on sides hi and jk options: both pairs of opposite sides are parallel; opposite angles are congruent; opposite sides are congruent; one pair of opposite sides is parallel and congruent

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# Brief Explanations:
To determine the test for parallelograms, we analyze the diagram. The marks on sides $ HI $ and $ JK $ indicate they are congruent, and the arrows (or the visual of the figure) suggest a pair of opposite sides is parallel and congruent? Wait, no—wait, the markings: the tick marks on $ HI $ and $ JK $ show those sides are congruent, and the direction (arrows) on $ HI $ and $ JK $ (if we assume the arrows mean parallel) but actually, the standard test: "One pair of opposite sides is parallel and congruent" is a test, but wait, looking at the options, the diagram has two sides (HI and JK) with congruency marks (the little ticks) and the other sides—wait, no, the correct test here: the figure shows one pair of opposite sides (HI and JK) with congruency marks (so congruent) and the direction (arrows) implying they are parallel? Wait, no, the options: let's recall the parallelogram tests. The test "One pair of opposite sides is parallel and congruent" is a valid test. Wait, but looking at the diagram, the sides $ HI $ and $ JK $ have the same number of tick marks (congruent) and the arrows (if we consider the direction) suggest they are parallel. Wait, no, maybe I misread. Wait, the options: "Opposite sides are congruent"—no, the diagram shows only one pair? Wait, no, the figure is a quadrilateral HIJK, with HI and JK marked as congruent (tick marks) and H and J with angles? Wait, no, the key is: the test "One pair of opposite sides is parallel and congruent" is a theorem. But wait, the diagram: the sides HI and JK have congruency marks (so congruent) and the arrows (direction) on HI and JK (if the arrows mean parallel) would mean one pair is parallel and congruent. But wait, maybe the correct option is "Opposite sides are congruent"? No, no—wait, the markings: in the diagram, HI and JK have the same tick marks (so they are congruent), and if we assume the other pair (HJ and IK) are... Wait, no, the options: let's check each option.

1. "Both pairs of opposite sides are parallel": the diagram doesn't show both pairs with parallel marks (only one pair maybe).
2. "Opposite angles are congruent": no angle markings.
3. "Opposite sides are congruent": the diagram shows only one pair (HI and JK) with congruency marks, not both pairs.
4. "One pair of opposite sides is parallel and congruent": the diagram has HI and JK with congruency marks (congruent) and the arrows (direction) suggesting they are parallel. So this test applies. Wait, but maybe I made a mistake. Wait, the standard: if one pair of opposite sides is both parallel and congruent, then it's a parallelogram. So the correct option is "One pair of opposite sides is parallel and congruent"? Wait, no, wait the diagram: the sides HI and JK have the same tick marks (congruent) and the arrows (parallel). So that's one pair. So the answer is the last option? Wait, no, maybe I messed up. Wait, let's re-express:

Looking at the diagram, sides $ HI $ and $ JK $ have congruency marks (so they are congruent) and the direction (arrows) implies they are parallel. So the test "One pair of opposite sides is parallel and congruent" is satisfied. But wait, the options: let's check again.

Wait, the options:

- Both pairs of opposite sides are parallel: needs two pairs, diagram doesn't show that.
- Opposite angles are congruent: no angle marks.
- Opposite sides are congruent: needs both pairs, diagram shows one pair.
- One pair of opposite sides is parallel and congruent: this is a valid test. So the correct option is "One pair of opposite sides is parallel and congruent"? Wait, no, maybe the diagram has HI and JK with congruency marks (so congruent) and the other sides... Wait, maybe the answer is "Opposite sides are congruent"? No, that test is "if both pairs of opposite sides are congruent, then it's a parallelogram". But the diagram only shows one pair. Wait, I think I made a mistake. Wait, the diagram: HI and JK have the same tick marks (congruent), and HJ and IK—wait, no, the figure is drawn as a parallelogram? Wait, no, the key is the markings. The tick marks on HI and JK mean they are congruent, and the arrows (direction) on HI and JK (if the arrows are parallel indicators) would mean they are parallel. So one pair is parallel and congruent. So the correct option is "One pair of opposite sides is parallel and congruent". Wait, but let's confirm the theorems. The theorem: If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. So that's the test. So the answer is the last option.
# Answer:
One pair of opposite sides is parallel and congruent (the last option in the list: "One pair of opposite sides is parallel and congruent")

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