Question

△fgi is equilateral and i is the midpoint of \\( \overline{hj} \\). complete the proof that \\( △fij≅△gih \\).

Explanation:

Step1: Recall properties of equilateral triangle

An equilateral triangle has all sides equal. So, for $\triangle FGI$, $\overline{FI}\cong\overline{GI}$ by the definition of an equilateral triangle.

Step2: Use congruence criteria

We have $\overline{IJ}\cong\overline{HI}$ (from the definition of mid - point, step 4), $\overline{FI}\cong\overline{GI}$ (from step 1), and $\overline{GH}\cong\overline{FJ}$ (given, step 3). By the Side - Side - Side (SSS) congruence criterion, if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.

Answer:

The reason for $\overline{FI}\cong\overline{GI}$ is "Definition of equilateral triangle".