The proof that HG ≅ EG is shown.

To determine the missing reason in the proof leading to the conclusion that triangles \( \triangle FEG \) and \( \triangle KHG \) are congruent, we need to analyze what information has been provided and what criteria for triangle congruence has been applied.

The proof provides several statements:

1. \( \angle EGF \cong \angle HGK \) (from vertical angles)
2. \( KH \parallel EF \) (given)
3. \( \angle F \cong \angle K \) (from alternate interior angles)
4. \( G \) is the midpoint of \( KF \) (given)
5. \( FG \cong KG \) (from the definition of midpoint)
6. \( \triangle FEG \cong \triangle KHG \) (missing reason)
7. \( HG \cong EG \) (by CPCTC)

To prove the triangles are congruent, we need to establish that there are enough corresponding angles and sides that fulfill one of the congruence criteria.

In this case, we have:

• Two pairs of angles: \( \angle EGF \cong \angle HGK \) and \( \angle F \cong \angle K \).
• One pair of sides: \( FG \cong KG \).

Here, we can apply the Angle-Angle-Side (AAS) criterion for triangle congruence because we have two pairs of angles and a corresponding non-included side. Therefore, the missing reason for step 6 in the proof is:

Reason: AAS (Angle-Angle-Side).

So the completed proof would look like this:

6. \( \triangle FEG \cong \triangle KHG \) 6. AAS