Question

Prove: HG ≅ EG

Triangles F E G and K H G are connected at point G.

What is the missing reason in the proof?

Statement
Reason
1. ∠EGF ≅ ∠HGK 1. vert. ∠s are ≅
2. KH ∥ EF 2. given
3. ∠F ≅ ∠K 3. alt. int. ∠s are ≅
4. G is the midpoint of KF 4. given
5. FG ≅ KG 5. def. of midpt.
6. △FEG ≅ △KHG 6. ?
7. HG ≅ EG 7. CPCTC
SAS
ASA
AAS
HL

Answer 1

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**.