In the given parallelogram $DEFG$, you are asked to identify which segment bisects $\overline{EG}$.

A key property of parallelograms is that the diagonals bisect each other. This means that if two diagonals intersect, each diagonal is split into two equal parts at the point of intersection.

In the diagram, $\overline{EG}$ and $\overline{DF}$ are the diagonals of parallelogram $DEFG$, and they intersect at point $H$. Since diagonals of a parallelogram bisect each other, it follows that $\overline{DF}$ bisects $\overline{EG}$ at point $H$.

Thus, the correct answer is: $\boxed{\overline{DF}}$

Would you like further clarification or details?

Here are 5 related questions to deepen understanding:

1. Why do diagonals in parallelograms bisect each other?
2. What other properties do diagonals of parallelograms have?
3. How can you prove that point $H$ is the midpoint of $\overline{EG}$?
4. What are the differences between the diagonals of a parallelogram and those of a rectangle?
5. Can you determine if $\triangle DHE$ is congruent to $\triangle FHG$?

Tip: In any parallelogram, the diagonals not only bisect each other but also divide the parallelogram into two congruent triangles.