Question

△efh is equilateral. complete the proof that △fgh≅△eih.
1 △efh is equilateral
2 ∠gfh≅∠hei
3 \overline{fg}≅\overline{ei}
4 \overline{eh}≅\overline{fh}
5 △fgh≅△eih

Explanation:

Step1: Given

\(\triangle EFH\) is equilateral (Given in the problem statement).

Step2: Given

\(\angle GFH\cong\angle HEI\) (Given in the problem's geometric figure).

Step3: Given

\(\overline{FG}\cong\overline{EI}\) (Given in the problem's geometric figure).

Step4: Property of equilateral triangle

Since \(\triangle EFH\) is equilateral, all its sides are equal. So, \(\overline{EH}\cong\overline{FH}\) (By the definition of an equilateral triangle: all sides are congruent).

Step5: SAS (Side - Angle - Side) congruence criterion

We have \(\overline{FG}\cong\overline{EI}\), \(\angle GFH\cong\angle HEI\), and \(\overline{FH}\cong\overline{EH}\). So, \(\triangle FGH\cong\triangle EIH\) by the SAS (Side - Angle - Side) congruence criterion.

Answer:

1. Given; 2. Given; 3. Given; 4. Definition of equilateral triangle; 5. SAS (Side - Angle - Side) congruence criterion.