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Math Help - parallelogram

  1. #1
    Member Veronica1999's Avatar
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    parallelogram

    If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent. What is the converse of this statement?
    Is it true?

    If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.

    This is false.

    Am I right?
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  2. #2
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    Hello, Veronica1999!

    If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent.
    What is the converse of this statement? .Is it true?

    If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.
    Yes, this is the converse.

    This is false. . No, it is true.

    We have quadrilateral \,ABCD with AB = CD,\;BC = AD.

    Draw diagonal \,AC.

    Let \angle 1 = \angle CAB,\;\angle 2 = \angle ACD.

    Code:
    
              D * - - - - - * C
               /      2  * /
              /       *   /
             /     *     /
            /   *       /
           / *  1      /
        A * - - - - - * B

    In triangles ABC and CDA\!:\;AB = CD,\;BC = AD,\;AC = AC.

    Hence: . \Delta ABC\,\cong\,\Delta CDA \quad\Rightarrow\quad \angle 1 \,=\,\angle 2

    Therefore: . AB \parallel CD


    Since AB = CD and AB \parallel CD,\:ABCD is a parallelogram.

    [Theorem: if two sides of a quadrilateral are equal and parallel,
    . . . . . . . . the quadrilateral is a parallelogram.]

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