Question

j is the midpoint of \\(\overline{hi}\\). complete the proof that \\(\angle h g j \cong \angle i g j\\).

statement	reason
2 \\(\overline{gh} \cong \overline{gi}\\)	given
3 \\(\overline{hj} \cong \overline{ij}\\)	definition of midpoint
4 \\(\overline{gj} \cong \overline{gj}\\)	reflexive property of congruence
5 \\(\triangle ghj \cong \triangle gij\\)
6 \\(\angle h g j \cong \angle i g j\\)	cpctc

Explanation:

Step1: Identify given info

We know $J$ is the midpoint of $\overline{HI}$ (Given), $\overline{GH} \cong \overline{GI}$ (Given).

Step2: Derive segment congruence

By definition of midpoint, $\overline{HJ} \cong \overline{IJ}$.

Step3: Use reflexive property

$\overline{GJ} \cong \overline{GJ}$ (Reflexive Property of Congruence).

Step4: Prove triangle congruence

Using Side-Side-Side (SSS) Congruence Postulate: since $\overline{GH} \cong \overline{GI}$, $\overline{HJ} \cong \overline{IJ}$, $\overline{GJ} \cong \overline{GJ}$, we conclude $\triangle GHJ \cong \triangle GIJ$.

Step5: Apply CPCTC

Corresponding Parts of Congruent Triangles are Congruent (CPCTC), so $\angle HGJ \cong \angle IGJ$.

Answer:

The completed proof has the missing reason for statement 5: Side-Side-Side (SSS) Congruence Postulate
The final proven statement is $\boldsymbol{\angle HGJ \cong \angle IGJ}$