Question

△fgi is equilateral. complete the proof that △fij ≅ △gih.

statement	reason
2 ∠hgi ≅ ∠ifj	given
3 fj ≅ gh	given
4 fi ≅ gi
5 △fij ≅ △gih

Explanation:

Step1: Determine the congruence criteria

Since $\triangle FGI$ is equilateral, \(FG = GI = FI\). Also, given \(FJ = GH\) (Statement 3), \(\angle HGI=\angle IFJ\) (Statement 2). We can use the Side - Angle - Side (SAS) congruence criterion.

Step2: Apply the SAS congruence

In \(\triangle FIJ\) and \(\triangle GIH\), we have \(FJ = GH\) (given), \(\angle IFJ=\angle HGI\) (given), and \(FI = GI\) (since \(\triangle FGI\) is equilateral, all its sides are equal).
By the SAS (Side - Angle - Side) congruence criterion, if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

Answer:

By the Side - Angle - Side (SAS) congruence criterion, \(\triangle FIJ\cong\triangle GIH\).