Question

△fgi is equilateral, \\(\overline{gh} \cong \overline{fj}\\), and \\(\angle ifj \cong \angle hgi\\). complete the proof that \\(\triangle fji \cong \triangle gih\\).

(image of a geometric figure with points j, i, f, h, g and a table with statement and reason columns for proof steps: 1. △fgi is equilateral; 2. \\(\overline{gh} \cong \overline{fj}\\); 3. \\(\angle ifj \cong \angle hgi\\); 4. \\(\overline{fi} \cong \overline{gi}\\); 5. \\(\triangle fji \cong \triangle gih\\))

Explanation:

Step1: Reason for Statement 1

Given that \(\triangle FGI\) is equilateral, by the definition of an equilateral triangle, all its sides are equal. So \(FI \cong GI\) (all sides of an equilateral triangle are congruent). The reason for statement 1 is "Given".

Step2: Reason for Statement 2

The statement \(\overline{GH} \cong \overline{FJ}\) is given, so the reason is "Given".

Step3: Reason for Statement 3

The statement \(\angle IFJ \cong \angle HGI\) is given, so the reason is "Given".

Step4: Reason for Statement 4

Since \(\triangle FGI\) is equilateral, \(FI = GI\) (sides of an equilateral triangle are congruent), so \(\overline{FI} \cong \overline{GI}\) by the definition of congruent segments (or "Sides of an equilateral triangle are congruent").

Step5: Reason for Statement 5

We have \(\overline{FJ} \cong \overline{GH}\) (statement 2), \(\angle IFJ \cong \angle HGI\) (statement 3), and \(\overline{FI} \cong \overline{GI}\) (statement 4). So by the Side - Angle - Side (SAS) congruence criterion, \(\triangle FIJ \cong \triangle GIH\). The SAS criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Here, \(FJ\) and \(FI\) with included angle \(\angle IFJ\) in \(\triangle FIJ\) are congruent to \(GH\) and \(GI\) with included angle \(\angle HGI\) in \(\triangle GIH\).

Answer:

1. Reason for Statement 1: Given
2. Reason for Statement 2: Given
3. Reason for Statement 3: Given
4. Reason for Statement 4: Sides of an equilateral triangle are congruent
5. Reason for Statement 5: SAS (Side - Angle - Side) Congruence Criterion