Question

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

Explanation:

Step1: Analyze the given information

We know that \(\angle GHD\) and \(\angle EDH\) are right - angles (\(\angle GHD=\angle EDH = 90^{\circ}\)), and \(GH\cong ED\). Also, from the vertical - angle theorem, \(\angle GFH=\angle EFD\) (vertical angles are equal). And \(GH\parallel ED\) (since the two right - angles are formed with the same line \(DH\)), so \(\angle HGF=\angle DEF\) (alternate interior angles).

Step2: Check the congruence criteria for each option

• Option 1: \(\triangle HFD\cong\triangle HFG\) by \(SSS\)

We do not have information about the equality of all three sides of \(\triangle HFD\) and \(\triangle HFG\).

• Option 2: \(\triangle FDH\cong\triangle FDE\) by \(ASA\)

We do not have information about the equality of two angles and the included side for \(\triangle FDH\) and \(\triangle FDE\) in the \(ASA\) sense.

• Option 3: \(\triangle GHF\cong\triangle EDF\) by \(SSS\)

We do not have information about the equality of all three sides of \(\triangle GHF\) and \(\triangle EDF\).

• Option 4: \(\triangle GHD\cong\triangle EDH\) by \(SAS\)

We have \(GH = ED\) (given), \(\angle GHD=\angle EDH=90^{\circ}\), and \(DH = DH\) (common side). So, by the \(SAS\) (Side - Angle - Side) congruence criterion, \(\triangle GHD\cong\triangle EDH\).

Answer:

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