To find the measures of the angles \( \angle QRS \) and \( \angle SRT \), we start with the information that these two angles are supplementary. This means that their measures add up to \( 180^\circ \).
Given that:
\[ m<QRS = x + 40 \] \[ m<SRT = x \]
Since the angles are supplementary, we can set up the following equation:
\[ m<QRS + m<SRT = 180 \]
Substituting the expressions for the angles into the equation gives:
\[ (x + 40) + x = 180 \]
Now simplify the equation:
\[ 2x + 40 = 180 \]
Next, subtract \( 40 \) from both sides:
\[ 2x = 140 \]
Now divide by \( 2 \):
\[ x = 70 \]
Now, we can find the measure of \( \angle QRS \):
\[ m<QRS = x + 40 = 70 + 40 = 110 \]
Thus, the measure of \( \angle QRS \) is \( \boxed{110^\circ} \).