Question

ng triangle congruence theorems
given \\( \angle g h d \\) and \\( \angle e d h \\) are right; \\( g h \cong e d \\)
which relationship in the diagram is true?
\\( \triangle f d h \cong \triangle f d e \\) by asa
\\( \triangle g h d \cong \triangle e d h \\) by sas
\\( \triangle h f d \cong \triangle h f g \\) by sss
\\( \triangle g h f \cong \triangle e d f \\) by sss

Explanation:

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

Given \(\angle GHD=\angle EDH = 90^{\circ}\), \(GH = ED\) (given), and \(HD=DH\) (common side).

Step2: Apply SAS (Side - Angle - Side) congruence criterion

In \(\triangle GHD\) and \(\triangle EDH\), we have two sides and the included right angle equal. For SAS, if in two triangles \(a = a'\), \(b=b'\) and the included angle \(\theta=\theta'\), then the triangles are congruent. Here \(a = GH\), \(a'=ED\), \(b = HD\), \(b'=DH\) and \(\angle GHD=\angle EDH\)

Answer:

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