Question

∠efj ≅ ∠gih, ∠ejf ≅ ∠hgi, and \\(\overline{ij} ≅ \overline{fg}\\). complete the proof that \\(\triangle efj ≅ \triangle hig\\).
image of two triangles hig and efj with i and f as intersection points

statement	reason
2. ∠ejf ≅ ∠hgi	given
3. \\(\overline{ij} ≅ \overline{fg}\\)	given
4. fj = ij + fi	additive property of length
5. gi = fg + fi	additive property of length
6. fj = fg + fi	substitution
7. gi = fj	transitive property of equality
8. blank for statement	blank for reason

Explanation:

Step1: Identify Congruence Criterion

We have two angles and a side. From steps 1, 2, and 7, we know $\angle EFJ \cong \angle GIH$, $\angle EJF \cong \angle HGI$, and $GJ = FJ$ (wait, correction: from step 7, $GI = FJ$). Wait, the angles are $\angle EFJ$ (angle at F in $\triangle EFJ$), $\angle GIH$ (angle at I in $\triangle HIG$), $\angle EJF$ (angle at J in $\triangle EFJ$), $\angle HGI$ (angle at G in $\triangle HIG$), and the side $FJ \cong GI$ (from step 7). So by AAS (Angle - Angle - Side) congruence criterion, $\triangle EFJ \cong \triangle HIG$.

Step2: Write the Congruence Statement

The statement for step 8 should be $\triangle EFJ \cong \triangle HIG$ and the reason is AAS (Angle - Angle - Side) Congruence Theorem, since we have two pairs of congruent angles and a pair of congruent sides (non - included, but AAS allows two angles and a non - included side).

Answer:

Statement: $\triangle EFJ \cong \triangle HIG$; Reason: AAS (Angle - Angle - Side) Congruence Theorem