Question

What are the steps for Given: △AFC≅△EFC, BC ≅ DC and ∠GHF≅∠GIF.
Prove: △AHB≅△EID.

Answer 1

I don’t have the diagram, and the proof depends on the exact positions of H, I, B, D, and G (for example whether H and I lie on AF and EF, whether B and D lie on AC and EC, how G is connected to the other points). Could you upload the figure or state where those points lie (e.g. “H is the intersection of GF with AB”, or “H lies on AF and I lies on EF”, etc.)?

Meanwhile, here is the standard outline (template) you can use once the incidence information is known. The target is to prove ΔAHB ≅ ΔEID; you will typically aim to establish two sides and the included angle (SAS) or two angles and a side (ASA).

Common useful facts you already have
- From ΔAFC ≅ ΔEFC: AF = EF, AC = EC, and corresponding angles (in particular ∠AFC = ∠EFC).
- From BC ≅ DC: CB = CD.
- From ∠GHF ≅ ∠GIF: an angle at H equals the corresponding angle at I (useful to match angles in triangles AHB and EID once you identify how GH/GI relate to AB/ED or to HF/IF).

Typical proof strategy (fill in depending on point positions)
1. Identify corresponding vertices: from the given congruence of AFC and EFC you will naturally match A ↔ E and then want to match H ↔ I and B ↔ D in the triangles to be proved congruent.
2. Show AH = EI.
- If H and I lie on AF and EF respectively, write AH = AF − FH and EI = EF − FI. Since AF = EF (from the congruence) it suffices to show FH = FI (so AH = EI).
- If FH = FI follows from the configuration (for example if G is chosen so that GF is the perpendicular bisector of AI, or if H and I are symmetric about F), mention that step and prove FH = FI from the given equal angle or symmetry.
3. Show HB = ID.
- If B and D lie on AC and EC respectively, HB = CB − HC and ID = CD − CI; since CB = CD (given) it suffices to show HC = CI (which you would get if H and I are chosen symmetrically relative to C or another bisector).
- Give the exact algebraic/subtractive argument once you know where B and D lie.
4. Show the included angles are equal.
- Use ∠GHF ≅ ∠GIF together with the incidence relations (for example if G, B, D are collinear or if GH is along HB and GI is along ID) to deduce ∠AHB = ∠EID (or the necessary corresponding angle equality).
5. Conclude ΔAHB ≅ ΔEID by SAS (or ASA, depending which equalities you established).

If you tell me the positions of H, I, B, D and G (or upload the diagram), I will convert the outline above into a complete step-by-step proof using the given facts.