Question

given: ∠ghd and ∠edh are right; \\(\overline{gh} \cong \overline{ed}\\)
which relationship in the diagram is true?
options:

• \\(\triangle ghd \cong \triangle edh\\) by sas
• \\(\triangle ghf \cong \triangle edf\\) by sss
• \\(\triangle fdh \cong \triangle fde\\) by asa
• \\(\triangle hfd \cong \triangle hfg\\) by sss

Explanation:

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

• Given \(\angle GHD=\angle EDH = 90^{\circ}\), \(GH = ED\) (given), and \(HD=DH\) (common side).
• By the SAS (Side - Angle - Side) congruence criterion: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
• In \(\triangle GHD\) and \(\triangle EDH\), \(GH = ED\), \(\angle GHD=\angle EDH\), \(HD = DH\).

Step2: Check other options

• For \(\triangle GHF\) and \(\triangle EDF\): We know \(GH = ED\), but we don't have information about all three sides (\(SSS\) requires three pairs of equal sides) to confirm congruence.
• For \(\triangle FDH\) and \(\triangle FDE\): \(\angle FDH

eq\angle FDE\) (one is part of a right - angle in the given figure), so \(ASA\) (Angle - Side - Angle) does not apply.

• For \(\triangle HFD\) and \(\triangle HFG\): We don't have information about all three sides (\(SSS\) requires three pairs of equal sides) to confirm congruence.

Answer:

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