To determine whether quadrilateral QRST is a parallelogram and to find the value of \( x \), we will first consider the properties of angles in a quadrilateral.
In any quadrilateral, the sum of the interior angles is \( 360^\circ \). We know that:
- \( \angle Q = 108^\circ \)
- \( \angle R = (108 - 2x)^\circ \)
Let’s denote the measures of angles \( S \) and \( T \) as \( \angle S \) and \( \angle T \) respectively. We will now apply the angle sum property of quadrilaterals:
\[ \angle Q + \angle R + \angle S + \angle T = 360^\circ \]
Substituting the known angles gives us:
\[ 108 + (108 - 2x) + \angle S + \angle T = 360 \]
Simplifying further:
\[ 216 - 2x + \angle S + \angle T = 360 \]
Rearranging the equation provides:
\[ \angle S + \angle T = 360 - 216 + 2x \] \[ \angle S + \angle T = 144 + 2x \]
Now, for QRST to be a parallelogram, the angles opposite each other must be equal, so we consider the angles Q and S, and R and T.
Let's say \( \angle S \) is equal to \( \angle Q \) and \( \angle T \) is equal to \( \angle R \):
If \( \angle S = \angle Q = 108^\circ \) (so its opposite angle is equal), then
\[ \angle T = (108 - 2x). \]
Since \( \angle T \) also equals \( R \):
\[ \angle T = 108 - 2x. \]
Setting these equal gives us:
\[ \angle T + \angle R = 180^\circ \quad \text{(since they should be supplementary)} \]
Thus,
\[ 108 + (108 - 2x) = 180 \] \[ 216 - 2x = 180 \] \[ -2x = 180 - 216 \] \[ -2x = -36 \] \[ x = 18. \]
Thus, we have found \( x = 18 \).
Now we assess our options considering Q and S as opposite angles:
Since QRST has opposite angles equal, we can conclude that quadrilateral QRST is indeed a parallelogram.
The correct response is:
"Quadrilateral QRST is a parallelogram because there are two pairs of parallel lines. The value \( x = 18 \) because the angles are supplementary."