When PQC is a right angle, 2x = 8-x and x = 8/3.

When ∠ QPC is a right angle, x=2(8-x) and x= 16/3.

(2) when AB//MP, x=2

When CD//MP, 8-x=2, x=6.

Therefore, when x=2, the quadrilateral ABPM is a parallelogram,

Or x=6, and the quadrilateral ABPM is a parallelogram.

(3) it does not exist.

Reason: S⊿MPQ= 1/2*(8-x)*(8-x)/2.

= 1/4 *(8-x)^2 (0≤X ≤8)

When x=0, the maximum value of S⊿MPQ is 16.

The half area of the isosceles trapezoid ABCD is1/2 * (4+8) * 4 √ 3 *1/2 =12 √ 3 >16.

So there is no such value of x, which makes the area of triangle PMQ half that of isosceles trapezoid ABCD.