Question

△fgi is equilateral and i is the midpoint of \\(\overline{hj}\\). complete the proof that \\(\triangle ghi \cong \triangle fji\\).
statement
1 \\(\triangle fgi\\) is equilateral
2 i is the midpoint of \\(\overline{hj}\\)
3 \\(\overline{fj} \cong \overline{gh}\\)
4 \\(\overline{ij} \cong \overline{hi}\\)
5 \\(\overline{hi} \cong \overline{gi}\\)
6 \\(\triangle ghi \cong \triangle fji\\)
reason

Explanation:

Step1: State given premise

Given (Problem statement)

Step2: State given premise

Given (Problem statement)

Step3: Use equilateral triangle property

All sides of an equilateral triangle are congruent, so $\overline{FJ} \cong \overline{GH}$ because $\triangle FGI$ is equilateral ($\overline{FG} \cong \overline{GI} \cong \overline{FI}$) and $\overline{FG} \cong \overline{GH}$ (marked congruent in diagram)

Step4: Use midpoint definition

Midpoint divides segment into 2 congruent parts: $\overline{IJ} \cong \overline{HI}$ since $I$ is midpoint of $\overline{HJ}$

Step5: Use equilateral triangle property

All sides of equilateral $\triangle FGI$ are congruent: $\overline{FI} \cong \overline{GI}$

Step6: Apply SSS congruence rule

$\triangle GHI \cong \triangle FJI$ by Side-Side-Side (SSS) congruence, as $\overline{GH} \cong \overline{FJ}$, $\overline{HI} \cong \overline{IJ}$, $\overline{GI} \cong \overline{FI}$

Answer:

Statement	Reason
$I$ is the midpoint of $\overline{HJ}$	Given
$\overline{FJ} \cong \overline{GH}$	Given (marked congruent) + Equilateral triangle side congruence
$\overline{IJ} \cong \overline{HI}$	Definition of midpoint
$\overline{FI} \cong \overline{GI}$	All sides of an equilateral triangle are congruent
$\triangle GHI \cong \triangle FJI$	Side-Side-Side (SSS) Triangle Congruence Postulate