To find \( m \angle IJM \), we can use the properties of parallel lines and transversals.
Given:
- Lines \( IK \) and \( LN \) are parallel.
- \( m \angle LMJ = 113^\circ \).
When two parallel lines are cut by a transversal, several angle relationships are created. Specifically, since \( IK \) and \( LN \) are parallel and \( LM \) acts as a transversal, we can identify the angles formed:
- \( m \angle LMJ \) and \( m \angle IJM \) are on the same side of the transversal and located internally. Since \( IK \) and \( LN \) are parallel, the two angles are supplementary.
Therefore, we can write the equation: \[ m \angle LMJ + m \angle IJM = 180^\circ. \]
We know that \( m \angle LMJ = 113^\circ \), so we can substitute this into the equation: \[ 113^\circ + m \angle IJM = 180^\circ. \]
Now, solving for \( m \angle IJM \): \[ m \angle IJM = 180^\circ - 113^\circ = 67^\circ. \]
Thus, the measure of \( \angle IJM \) is \( \boxed{67^\circ} \).