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# What must one justify to prove that a quadrilateral ABCD is a parallelogram?

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Regents Prep explains that the quickest method for proving that a quadrilateral is a parallelogram is if one pair of opposite sides of the quadrilateral are both parallel and congruent. However, several other forms of proof exist as well.

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According to Wikipedia, in Euclidean geometry, a parallelogram is a simple quadrilateral, with four edges and four corners, characterized by its two pairs of parallel sides. Because of its parallel orientation, the opposite sides of a parallelogram are equal in length. The opposite angles of a parallelogram are also equal in measure. This congruence is proven by Euclid's parallel postulate, also known as Euclid's fifth postulate; it is part of Euclid's "Elements," which serves as the basis for the shortest method of proof previously described.

The basic definition of a parallelogram lends itself to several other methods of proof. For instance, one can prove that a quadrilateral is a parallelogram if the two pairs of opposite angles in the quadrilateral are congruent. Further, if the consecutive angles in the quadrilateral are supplementary, or equal to 180 degrees, then it is a parallelogram. Finally, one can also prove a quadrilateral is a parallelogram if the two diagonals of the quadrilateral can bisect each other, or divide each other equally in half.

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## Related Questions

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One proof regarding a parallelogram is used to show that the two pairs of opposite sides are congruent. When given a parallelogram with angles A, B, C and D, the line segments AB and CD are congruent, as are the line segments BC and AD.

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A rhombus is a four-sided figure with opposite sides parallel, making it a parallelogram. The rhombus differs from other parallelograms in that all four sides are equal in length.

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To calculate the area of a parallelogram from vectors, find the cross product of the sides and vectors. The area equals the length of the cross product of two vectors.