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Parallelograms - Proving a Quadrilateral is a Parallelogram I. The definition of a parallelogram states that if a quadrilateral is a parallelogram, then

opposite sides are parallel. Remember, however, that definitions can also be stated the other way around. In other words, the definition of a parallelogram also states that if both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram. For example, in the diagram below, if ABDC and ADBC , then quadrilateral ABCD is a parallelogram.

II. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. For example, in the diagram below, if AB DC and AD BC , then quadrilateral ABCD is a parallelogram.

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continued on next page III. If one pair of opposite sides of a parallelogram is both congruent and parallel,

then the quadrilateral is a parallelogram. For example, in the diagram below, if AB DC and ABDC , then quadrilateral ABCD is a parallelogram.

IV. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. For example, in the diagram below, if A C and B D, then quadrilateral ABCD is a parallelogram.

V. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. For example, in the diagram below, if AX XC and DX XB , then quadrilateral ABCD is a parallelogram.

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