Question

δefh is equilateral. complete the proof that δehi ≅ δfhg.
1 δefh is equilateral
2 ∠gfh ≅ ∠hei
3 \overline{ei} ≅ \overline{fg}
4 \overline{eh} ≅ \overline{fh}
5 δehi ≅ δfhg

Explanation:

Step1: Given

Given that \(\triangle EFH\) is equilateral.

Step2: Given

Given that \(\angle GFH\cong\angle HEI\).

Step3: Given

Given that \(\overline{EI}\cong\overline{FG}\).

Step4: Definition of equilateral triangle

Since \(\triangle EFH\) is equilateral, all its sides are equal. So, \(\overline{EH}\cong\overline{FH}\).

Step5: SAS (Side - Angle - Side) Congruence Criterion

In \(\triangle EHI\) and \(\triangle FHG\), we have \(\overline{EI}\cong\overline{FG}\) (Step 3), \(\angle HEI\cong\angle GFH\) (Step 2), and \(\overline{EH}\cong\overline{FH}\) (Step 4). By the SAS congruence criterion, \(\triangle EHI\cong\triangle FHG\).

Answer:

The proof is completed using the SAS (Side - Angle - Side) congruence criterion.