Prove: HG ≅ EG

To determine the missing reason for step 6 in the proof, let's analyze the information given:

1. We know that \( \triangle FEG \) and \( \triangle KHG \) are connected at point \( G \).
2. From the statements:
   • \( \angle EGF \cong \angle HGK \) by vertical angles.
   • \( KH \parallel EF \), which gives alternate interior angles \( \angle F \cong \angle K \).
   • \( G \) is the midpoint of \( KF \), meaning that \( FG \cong KG \) (by the definition of a midpoint).

Now, we can summarize the information:

• We have:
  • Two angles \( \angle EGF \cong \angle HGK \) (vertical angles).
  • One pair of angles \( \angle F \cong \angle K \) (alternate interior angles).
  • One pair of sides \( FG \cong KG \) (from the definition of midpoint).

Thus, we can use the ASA (Angle-Side-Angle) postulate, because we have:

• \( \angle F \cong \angle K \) (one pair of angles),
• \( FG \cong KG \) (the side between those angles),
• \( \angle EGF \cong \angle HGK \) (another pair of angles).

Therefore, the reason to support the congruence of the triangles \( \triangle FEG \) and \( \triangle KHG \) is:

• ASA (Angle-Side-Angle).

So the missing reason is ASA.