1. ## isosceles trapezoid

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

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

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

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

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

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

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

Therefore, $MNOA$ is an isosceles trapezoid.