To prove that quadrilateral ABCD is a parallelogram, we can use the properties that define a parallelogram. A quadrilateral is classified as a parallelogram if at least one of the following conditions holds:
- Opposite sides are congruent.
- Opposite angles are congruent.
- One pair of opposite sides is both congruent and parallel.
- The diagonals bisect each other.
Now, let's evaluate the options given:
a. Adjacent sides of the figure are congruent: This condition is not sufficient to prove that quadrilateral ABCD is a parallelogram. Congruent adjacent sides may form a triangle or other shapes, not necessarily a parallelogram.
b. Adjacent angles of the figure are congruent: Similar to option (a), this does not provide sufficient information to conclude that ABCD is a parallelogram. It could describe other shapes, like a rectangle or rhombus, but not necessarily a parallelogram.
c. Opposite sides of the figure are congruent: If opposite sides are congruent (i.e., \( AB \cong CD \) and \( AD \cong BC \)), then according to the properties of parallelograms, ABCD is indeed a parallelogram.
d. Adjacent sides of the figure are parallel: If adjacent sides of the quadrilateral are parallel, this does not follow the definition that allows concluding that ABCD is a parallelogram. Only parallel opposite sides are critical in determining if a figure is a parallelogram.
Conclusion:
The correct choice that proves quadrilateral ABCD is a parallelogram is:
c. Opposite sides of the figure are congruent.
This is sufficient to conclude that ABCD is a parallelogram according to the properties that define parallelograms.
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