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Thread: isosceles trapezoid

  1. #1
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    isosceles trapezoid

    Could someone please help me prove that this is an isosceles trapezoid? Thank you in advance.
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  2. #2
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    Hello, sanee66!

    In quadrilateral MNOA, triangles PNO and PMA are congruent
    and APO is an isosceles triangle.
    Prove that MNOA is an isosceles triangle.
    Code:
                          P
              M * - - - - * - - - - * N
               /     β  *   *  β     \
              /      *        *       \
             /    *              *     \
            /   *                   *   \
           / *  α                   α  * \
        A * - - - - - - - - - - - - - - - * O

    Since $\displaystyle \Delta PNO \cong \Delta PMA\!:\;MA = NO$
    . . So we must prove that: .$\displaystyle MN \parallel AO$

    Since $\displaystyle \Delta PNO \cong \Delta PMA\!:\;\angle MPA = \angle NPO$
    . . Let $\displaystyle \angle MPA = \angle NOP = \beta$

    We note that: .$\displaystyle 2\beta + \angle APO \:=\:180^o$ .[1]


    Since $\displaystyle \Delta APO$ is isosceles, $\displaystyle \angle PAO = \angle POA$
    . . Let $\displaystyle \angle PAO = \angle POA = \alpha$

    In $\displaystyle \Delta APO\!:\;2\alpha + \angle APO \:=\:180^o\quad\Rightarrow\quad \angle APO \:=\:180^o -2\alpha$

    Substitute into [1]: .$\displaystyle 2\beta + (180^o -2\alpha) \:=\:180^o\quad\Rightarrow\quad \alpha = \beta$

    Hence: .$\displaystyle MN \parallel AO$ .(alternate-interior angles)

    Therefore, $\displaystyle MNOA$ is an isosceles trapezoid.

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