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Math Help - Converses help?

  1. #1
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    Converses help?

    If the quadrilateral is an isosceles trapezoid, then a pair of base angles are congruent.

    If the quadrilateral is a kite, then the nonvertex angles are congruent.

    If the quadrilateral is a kite, then the major diagonal bisects the vertex angles.

    I'm really confused about the converses. I got false for all of them?
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  2. #2
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    Quote Originally Posted by Rinnie View Post
    If the quadrilateral is an isosceles trapezoid, then a pair of base angles are congruent.

    If the quadrilateral is a kite, then the nonvertex angles are congruent.

    If the quadrilateral is a kite, then the major diagonal bisects the vertex angles.

    I'm really confused about the converses. I got false for all of them?
    It would help if you told us WHAT the question was for which you got "false for all of them"! Is it whether or not the converse to the statement is true or false? What is the converse of each of these statements?
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  3. #3
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    It's whether the converses are true or false. I'm having trouble finding the converses.
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  4. #4
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    Hello, Rinnie!

    The last one is true.

    First, we write the convese and see if it is also true,
    . . or try to find a counterexample.


    If the quadrilateral is an isosceles trapezoid,
    . . then a pair of base angles are congruent.
    Converse:
    If a quadrilateral has a pair of equal base angles, it is an isosceles trapezoid.

    Code:
                              *
                          *    *
                      *         *
                  *              *
              *                   *
             *                     *
            *                       *
           * θ                     θ *
          *   *   *   *   *   *   *   *

    This is a quarilateral with equal base angles,
    . . but it is not an isosceles trapezoid.

    The converse is false.




    If the quadrilateral is a kite,
    . . then the nonvertex angles are congruent.
    Converse
    If the nonvertex angles of a quadrilateral are congruent,
    . . the quadrilateral is a kite.
    Code:
                *
               *  *
              *     *
             *        *
            * θ         *
              *           *
                *         θ *
                  *        *
                    *     *
                      *  *
                        *

    This is a quadrilateral with congruent nonvertex angles,
    . . but it is not a kite.

    The converse is false.




    If the quadrilateral is a kite,
    . . then the major diagonal bisects the vertex angles.
    Converse:
    If the major diagonal of a quadrilateral bisects the vertex angles,
    . . then the quadrilateral is a kite.
    Code:
                    A
                    *
                   *|*
                  * | *
                 *α | α*
                *   |   *
               *    |    *
            B *     |     * D
                * β | β *
                  * | *
                    *
                    C
    There is no way to avoid drawing a kite.

    We have: . \begin{Bmatrix}\angle BAC \:=\: \angle DAC \:=\: \alpha \\ AC \:=\:AC \\\angle BCA \:=\:\angle DCA \:=\:\beta \\ \therefore \Delta ABC \cong \Delta ADC \\ (a.s.a) \end{Bmatrix}

    Therefore, ABCD is a kite.


    The converse is true.

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