To determine the measure of angle \( m\angle QPS \), we can use the properties of parallel lines and the angles formed by a transversal.
Since \( OQ \) and \( RT \) are parallel lines and \( PS \) is a transversal that intersects them at points \( S \) and \( Q \), we can analyze the angles.
Given that \( m\angle TSP = 138^\circ \):
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\( m\angle TSP \) and \( m\angle QPS \) are same-side interior angles because they are both located between the parallel lines \( OQ \) and \( RT \).
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The sum of the measures of same-side interior angles is \( 180^\circ \).
Thus, we can find \( m\angle QPS \) as follows:
\[ m\angle QPS + m\angle TSP = 180^\circ \]
Substituting the value of \( m\angle TSP \):
\[ m\angle QPS + 138^\circ = 180^\circ \]
Now, isolate \( m\angle QPS \):
\[ m\angle QPS = 180^\circ - 138^\circ \] \[ m\angle QPS = 42^\circ \]
Therefore, \( m\angle QPS = 42^\circ\).