If \( DF \) and \( GI \) are parallel lines, and \( \angle IHE \) (which I'm assuming corresponds to \( mIHE \)) is \( 118^\circ \), we can use the properties of parallel lines and transversals to find \( mDEH \).
Since \( DF \) and \( GI \) are parallel, \( \angle IHE \) is an exterior angle formed by a transversal intersecting the parallel lines. The angle \( mDEH \) will be an interior angle located on the same side of the transversal as \( mIHE \).
According to the properties of parallel lines, the interior angles on the same side of the transversal are supplementary. This means:
\[ mIHE + mDEH = 180^\circ \]
Substituting the known value:
\[ 118^\circ + mDEH = 180^\circ \]
Now, solving for \( mDEH \):
\[ mDEH = 180^\circ - 118^\circ = 62^\circ \]
Thus, the measure of angle \( DEH \) is
\[ \boxed{62^\circ} \]
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