Refrection of a function across the y-axis, changes the sign of x in the function.
Thus, given the function:

Refrection of the function across the y-axis will result in the function:

Refrecting a function across the x-axis, changes the sign of the function.
Thus, refrecting the function:

across the x-axis will result in the function:

The graph of

and

is attached.
The green curve represents the graph of the function

, while the orange curve represents the graph of the function

.
Since measure REU equal measure SFT, RE=FT and SF=EU then the two triangles REU et SFT are similar.
Then we deduce that the two sides RU and ST are equal, RU=ST.
Also, since the two triangles above are similar, then the two angles FST and RUE are equal. We deduce that the two lines RU and ST are parallel (interior opposite angles principles.)
We have two facts now:
RU = ST and RU parallel to ST, we deduce that the quadrilateral is a parallelogram.