In parallelogram DEFG, the segment that bisects the line segment EG is the segment EH.
In a parallelogram, the diagonals bisect each other at their intersection point, which in this case is point H. The line segment EH connects point E (top right corner) to point H (the intersection of the diagonals) and extends vertically to the midpoint of segment EG, thereby bisecting it.
Therefore, the correct answer is EH.