To solve for the measure of angle \( mFGJ \), we can use the information provided about the parallel lines \( FH \) and \( IK \).
Since \( FH \) and \( IK \) are parallel lines, angle \( IJG \) and angle \( FGJ \) are corresponding angles. This means they are equal in measure.
Given that \( mIJG = 110^\circ \):
\[ mFGJ = mIJG = 110^\circ \]
Thus, the measure of angle \( FGJ \) is \( 110^\circ \).