Question

g is the midpoint of \\( \overline{fh} \\), \\( \overline{hj} \cong \overline{fi} \\), and \\( \overline{gj} \cong \overline{gi} \\). complete the proof that \\( \triangle ghj \cong \triangle gfi \\).

Explanation:

Step1: Midpoint definition

Since \(G\) is the midpoint of \(\overline{FH}\), by the definition of a midpoint, \(FG = GH\) (i.e., \(\overline{FG}\cong\overline{GH}\)).

Step2: SSS congruence criterion

We know that \(\overline{HJ}\cong\overline{FI}\), \(\overline{GJ}\cong\overline{GI}\), and \(\overline{FG}\cong\overline{GH}\).
In \(\triangle GHJ\) and \(\triangle GFI\), we have three pairs of congruent sides.
By the Side - Side - Side (SSS) congruence criterion for triangles: If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Answer:

1. Reason: Definition of midpoint
2. Reason: SSS (Side - Side - Side) congruence criterion.