Question

1 j is the midpoint of \\( \overline { f h } \\) given
2 j is the midpoint of \\( \overline { g i } \\) given
3 \\( \overline { g h } \cong \overline { f i } \\) given
4 \\( \overline { f j } \cong \overline { h j } \\)
5 \\( \overline { g j } \cong \overline { i j } \\) definition of midpoint
6 \\( \delta f i j \cong \delta h g j \\)

Explanation:

Step1: Definition of midpoint

Since \(J\) is the midpoint of \(\overline{FH}\), by the definition of midpoint, if a point \(J\) is the midpoint of a line segment \(\overline{FH}\), then \(FJ = JH\), which means \(\overline{FJ}\cong\overline{HJ}\).

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

We know that \(\overline{FJ}\cong\overline{HJ}\) (from Step 1), \(\overline{GJ}\cong\overline{IJ}\) (from the definition of midpoint of \(\overline{GI}\) as \(J\) is the midpoint of \(\overline{GI}\)), and \(\angle FJI=\angle HJG\) (vertically - opposite angles are equal).
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. So, \(\triangle FIJ\cong\triangle H GJ\)

Answer:

1. Definition of midpoint; 6. SAS (Side - Angle - Side) congruence criterion.