To prove that quadrilateral \( ABCD \) is a parallelogram, we can use one of the established properties of parallelograms. Let's examine the options provided:
A: Adjacent angles of the figure are congruent.
This condition does not prove that a quadrilateral is a parallelogram because having congruent adjacent angles does not imply parallelism or equal lengths of sides.
B: Opposite sides of the figure are congruent.
This is a valid condition. If both pairs of opposite sides are equal in length (i.e., \( AB = CD \) and \( AD = BC \)), then quadrilateral \( ABCD \) can be concluded as a parallelogram by the definition of a parallelogram.
C: Adjacent sides of the figure are congruent.
This condition also does not imply that the quadrilateral is a parallelogram, as congruent adjacent sides do not guarantee that opposite sides are equal or that any sides are parallel.
D: Adjacent sides of the figure are parallel.
If adjacent sides are parallel, it would imply that the quadrilateral might be a rectangle or another type of quadrilateral, rather than necessarily being a parallelogram, unless both pairs of opposite sides are shown to be parallel.
Conclusion: The strongest and most applicable option that directly shows that quadrilateral \( ABCD \) is a parallelogram is B: opposite sides of the figure are congruent. Thus, if \( AB = CD \) and \( AD = BC \), we can conclude that quadrilateral \( ABCD \) is a parallelogram.
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