Given: ∠GHD and ∠EDH are right; GH ≅ ED

To determine which relationship in the triangles is true given the conditions:

1. Triangles GHD and EDH overlap and intersect at point F.
2. Triangles share side HD.
3. Sides GH and ED are congruent (GH ≅ ED).
4. Angles GHD and EDH are right angles (both are 90 degrees).

We can analyze each option:

1. △GHD ≅ △EDH by SAS:

   • The two triangles have a side (HD) in common, and both triangles have a right angle at H and D. Additionally, we have GH ≅ ED. Thus, we have:
     • One side (HD) is common,
     • The included angle (GHD and EDH, which are both right angles) is congruent,
     • The other side (GH and ED) are congruent.
   • Therefore, we can conclude that the two triangles are congruent by SAS.

2. △GHF ≅ △EDF by SSS:

   • This assertion requires us to know the lengths of all sides (GH, HF, DE, EF, DF) to claim that they are congruent. The information provided doesn't specify these lengths definitively for triangles GHF and EDF. Thus, we cannot confirm this statement based on the given information.

3. △FDH ≅ △FDE by ASA:

   • For this relationship to be true, we would require two angles and the included side to be congruent between the two triangles. The information does not explicitly provide us with enough angle and side relationships for FDH and FDE to conclude congruence by ASA.

4. △HFD ≅ △HFG by SSS:

   • Similar to the previous situation, for this to hold true, we need to know the lengths of all three sides definitively. We do not have sufficient information to confirm the lengths of sides HF, HG, DF, or DH, so we cannot verify this assertion as true.

Therefore, the only valid congruence relationship based on the conditions given is:

△GHD ≅ △EDH by SAS.