Question

g is the midpoint of \\( \overline { f h } \\). complete the proof that \\( \triangle g h j \cong \triangle g f i \\).
1 g is the midpoint of \\( \overline { f h } \\)
2 \\( \overline { h j } \cong \overline { f i } \\)
3 \\( \overline { g i } \cong \overline { g j } \\)
4 \\( \overline { f g } \cong \overline { g h } \\)
5 \\( \triangle g h j \cong \triangle g f i \\)

Explanation:

Step1: Given information

Statement 1: \(G\) is the midpoint of \(\overline{FH}\). Reason: Given.

Step2: Congruent segments from the figure

Statement 2: \(\overline{HJ}\cong\overline{FI}\). Reason: Given (marked in the figure).

Step3: Congruent segments from the figure

Statement 3: \(\overline{GI}\cong\overline{GJ}\). Reason: Given (marked in the figure).

Step4: Mid - point definition

Since \(G\) is the midpoint of \(\overline{FH}\), by the definition of a mid - point (\(FG = GH\)), so \(\overline{FG}\cong\overline{GH}\). Reason: Definition of mid - point.

Step5: SSS (Side - Side - Side) congruence criterion

In \(\triangle GHJ\) and \(\triangle GFI\), we have \(\overline{HJ}\cong\overline{FI}\), \(\overline{GI}\cong\overline{GJ}\), \(\overline{FG}\cong\overline{GH}\). So, \(\triangle GHJ\cong\triangle GFI\) by the SSS (Side - Side - Side) congruence criterion. Reason: SSS Congruence Theorem.

Answer:

1. Given; 2. Given; 3. Given; 4. Definition of mid - point; 5. SSS Congruence Theorem.