Question

given: ∠ghd and ∠edh are right; (overline{gh}congoverline{ed}) which relationship in the diagram is true? (\triangle ghdcong\triangle edh) by sas (\triangle ghfcong\triangle edf) by sss (\triangle fdhcong\triangle fde) by asa (\triangle hfdcong\triangle hfg) by sss

Explanation:

Step1: Analyze \(\triangle GHD\) and \(\triangle EDH\)

• Given \(\angle GHD=\angle EDH = 90^{\circ}\) (right angles).
• Given \(GH\cong ED\).
• \(HD = DH\) (common side).
• By the Side - Angle - Side (SAS) congruence criterion (\(GH\cong ED\), \(\angle GHD\cong\angle EDH\), \(HD\cong DH\)), \(\triangle GHD\cong\triangle EDH\).

Step2: Analyze other options

• For \(\triangle GHF\cong\triangle EDF\) by SSS: We know \(GH\cong ED\), but we need to check \(GF = EF\) and \(HF=DF\). There is no information given initially about \(GF = EF\) or \(HF = DF\) (except what can be derived from \(\triangle GHD\cong\triangle EDH\) which is not relevant for SSS here).
• For \(\triangle FDH\cong\triangle FDE\) by ASA: \(\angle FDH

eq\angle FDE\) ( \(\angle EDH = 90^{\circ}\) and \(\angle FDH\) is part of \(\angle EDH\) but not equal to \(\angle FDE\) as per the given diagram).

• For \(\triangle HFD\cong\triangle HFG\) by SSS: There is no information given about \(FD = FG\) or \(HF\) being a common side in a way that would satisfy SSS (since we don't know about the third - side lengths).

Answer:

\(\triangle GHD\cong\triangle EDH\) by \(SAS\)