Question

1 (\triangle fph) is equilateral given
2 (overline{fg}congoverline{ei}) given
3 (angle heicongangle gfh) given
4 (overline{eh}congoverline{fh})
5 (\triangle fehcong\triangle fhg)

Explanation:

Step1: Properties of equilateral triangle

Since \(\triangle EFH\) is equilateral, by the property of an equilateral triangle, all its sides are equal. So, \(\overline{EH}=\overline{FH}\) (sides of an equilateral triangle are equal).

Step2: Using the SAS (Side - Angle - Side) congruence criterion

We have \(\overline{EI}=\overline{FG}\) (given), \(\angle HEI=\angle GFH\) (given), and \(\overline{EH}=\overline{FH}\) (from step 1). By the SAS congruence criterion (\(SAS\): 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), \(\triangle EHI\cong\triangle FHG\)

Answer:

1. Sides of an equilateral triangle are equal; 5. \(SAS\) (Side - Angle - Side) congruence criterion.