To determine the measure of angle \( m∠2 \), we can use the fact that angles on a straight line sum to \( 180^\circ \).
From the diagram, if we denote the angles as follows:
- \( m∠4 = 31^\circ \)
- \( m∠2 \) and \( m∠4 \) are on the straight line along with \( m∠1 \) and \( m∠5 \).
Since angle \( m∠4 \) is vertically opposite to the region of angles around point \( * \) (the intersection), we have: \[ m∠2 + m∠4 = 180^\circ \]
Substituting the known value for \( m∠4 \): \[ m∠2 + 31^\circ = 180^\circ \]
Now, solve for \( m∠2 \): \[ m∠2 = 180^\circ - 31^\circ \] \[ m∠2 = 149^\circ \]
Thus, the measure of angle \( m∠2 \) is \( 149^\circ \).