To find the value of \( m \angle POM \), we will need to analyze the situation in which \( NP \) and \( QS \) are parallel lines, and \( m \angle SRT = 112^\circ \).
Since \( NP \) and \( QS \) are parallel lines, we can apply the concept of alternate interior angles. The angles \( SRT \) and \( POM \) are corresponding angles. Therefore, if one angle \( SRT \) measures \( 112^\circ \), the corresponding angle \( POM \) will also measure the same.
Thus, \[ m \angle POM = m \angle SRT = 112^\circ \]
The final answer is \( \boxed{112^\circ} \).