Question
Prove that quadrilateral ABCD is a parallelogram. How do you know that the figure is a parallelogram?
A: Adjacent angles of the figure are congruent.
B: opposite sides of the figure are congruent.
C: adjacent sides of the figure are congruent.
D: Adjacent sides of the figure are parallel.
A: Adjacent angles of the figure are congruent.
B: opposite sides of the figure are congruent.
C: adjacent sides of the figure are congruent.
D: Adjacent sides of the figure are parallel.
Answers
GPT-4o mini
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.
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.