Given the information that \( NP \) and \( QS \) are parallel lines and the angle \( m \angle NOM = 69^\circ \), we want to find the measure of angle \( m \angle POM \).
To solve this, we can use the concept of alternate interior angles. When two parallel lines are crossed by a transversal, the pairs of alternate interior angles are equal.
Since \( NP \) and \( QS \) are parallel and \( OM \) acts as a transversal, the angle \( m \angle NOM \) and angle \( m \angle POM \) are alternate interior angles. Therefore, we have:
\[ m \angle POM = m \angle NOM \]
Substituting the value of \( m \angle NOM \):
\[ m \angle POM = 69^\circ \]
Thus, the measure of angle \( m \angle POM \) is \( \boxed{69^\circ} \).
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