Question

given: ∠ghd and ∠edh are right; \\(\overline{gh} \cong \overline{ed}\\)
which relationship in the diagram is true?
\\(\bigcirc\\) \\(\triangle ghd \cong \triangle edh\\) by sas
\\(\bigcirc\\) \\(\triangle ghf \cong \triangle edf\\) by sss
\\(\bigcirc\\) \\(\triangle fdh \cong \triangle fde\\) by asa
\\(\bigcirc\\) \\(\triangle hfd \cong \triangle hfg\\) by sss

Explanation:

Step1: Analyze △GHD and △EDH

∠GHD = ∠EDH = 90° (right angles), $\overline{GH} \cong \overline{ED}$ (given), $\overline{HD} \cong \overline{DH}$ (common side).
Check congruence: Two sides (GH=ED, HD=DH) and included angle (right angle) → SAS.

Step2: Verify other options

• △GHF ≅ △EDF by SSS: Only GH=ED is given; GF/EF, HF/DF unknown → invalid.
• △FDH ≅ △FDE by ASA: No info on angles/sides for ASA → invalid.
• △HFD ≅ △HFG by SSS: HF common, but FD/FG, HD/HG unknown → invalid.

Answer:

A. △GHD ≅ △EDH by SAS