Question

What are the steps for Given: △ABH≅△EDI and GH ≅ GI. Prove: ∠HGF≅∠IGF.

Answer 1

I don’t have the diagram, so I’m missing where G and F lie relative to A,B,D,E,H,I. Could you upload or describe the figure (for example: are F and G on BD, is F the midpoint of HI, is GF a perpendicular bisector, etc.)?

Meanwhile here are the plausible proof routes and a full proof in the most common interpretation:

Two standard ways to get ∠HGF ≅ ∠IGF are
- show GF is the perpendicular bisector of HI (then GF bisects ∠HGI), or
- show triangles GHF and GIF are congruent (then the base angles at G are equal).

Both approaches need one more relation besides GH ≅ GI; that extra relation must come from the congruence △ABH ≅ △EDI (how the labelled vertices correspond depends on the diagram).

Common assumed configuration (used in many text problems)
- F is the midpoint of HI (so HF = IF),
- G is some point with GH = GI (given).

Under that assumption a direct proof:

1. Given GH = GI (given).
2. Given HF = IF (F is midpoint of HI — this must be given or deduced from the diagram).
3. GF is common to triangles GHF and GIF.
4. Therefore in triangles GHF and GIF we have GH = GI, HF = IF, GF = GF, so the triangles are congruent by SSS.
5. Corresponding angles ∠HGF and ∠IGF are equal by CPCTC (corresponding parts of congruent triangles are congruent).
6. Hence ∠HGF ≅ ∠IGF. QED.

If instead F is the foot of the perpendicular from G to HI (so GF ⟂ HI), then GH and GI are symmetric about GF (because G lies on the perpendicular bisector if additionally HF = IF), and a similar argument using right triangles or reflection gives the angle equality.

If you post or describe the actual figure (where F and G are located and any additional givens like “F is midpoint of HI” or “GF ⟂ HI” or “F lies on BD”), I will give a precise step-by-step proof that uses the given congruence △ABH ≅ △EDI and the given GH ≅ GI.