Question

The proof that HG ≅ EG is shown.

Given: G is the midpoint of KF

KH ∥ EF

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

The missing reason is ASA.

Justification: ∠EGF ≅ ∠HGK (vertical angles), ∠EFG ≅ ∠HKG (alternate interior angles since KH ∥ EF), and FG ≅ KG (G is midpoint of KF). So the two triangles have two angles and the included side equal, hence △FEG ≅ △KHG by ASA. Then HG ≅ EG by CPCTC.