Question

δfgi is equilateral. complete the proof that δfij ≅ δgih.
1 δfgi is equilateral given
2 \\( \overline { g h } \cong \overline { f j } \\) given
3 \\( \angle i f j \cong \angle h g i \\) given
4 \\( \overline { f i } \cong \overline { g i } \\)
5 \\( \delta f i j \cong \delta g i h \\)

Explanation:

Step1: Property of equilateral triangle

Since \(\triangle FGI\) is equilateral, all its sides are equal. So, \(\overline{FI}\cong\overline{GI}\) (by the definition of an equilateral triangle: all sides of an equilateral triangle are congruent).

Step2: Use of SAS (Side - Angle - Side) congruence criterion

We have \(\overline{FI}\cong\overline{GI}\) (from step 1), \(\overline{GH}\cong\overline{FJ}\) (given), and \(\angle IFJ\cong\angle HGI\) (given).
By the SAS (Side - Angle - Side) congruence criterion: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
For \(\triangle FIJ\) and \(\triangle GIH\), in \(\triangle FIJ\) we have sides \(\overline{FJ}\), \(\overline{FI}\) and included angle \(\angle IFJ\); in \(\triangle GIH\) we have sides \(\overline{GH}\), \(\overline{GI}\) and included angle \(\angle HGI\).

Answer:

1. Definition of equilateral triangle; 5. SAS (Side - Angle - Side)