Question

1 agik is equilateral 2 ∠ikj ≅ ∠hgi 3 ∠jik ≅ ∠gih 4 overline{ik} ≅ overline{gi} 5 △ijk ≅ △ihg

Explanation:

Step1: Definition of equilateral triangle

Since \(\triangle GIK\) is equilateral, by the definition of an equilateral triangle, all its sides are equal. So \(GI = IK=GK\).

Step2: Vertical angles

\(\angle IKJ\) and \(\angle HGI\) are vertical angles. By the vertical - angles theorem, \(\angle IKJ\cong\angle HGI\).

Step3: Alternate - interior angles

If we consider the lines (not explicitly named but from the figure's structure), \(\angle JIK\) and \(\angle GIH\) are alternate - interior angles. So \(\angle JIK\cong\angle GIH\) (alternate - interior angles theorem).

Step4: Side - Angle - Side (SAS) congruence

We know that \(IK\cong GI\) (from the equilateral triangle \(\triangle GIK\)), \(\angle IKJ\cong\angle HGI\) (vertical angles), \(\angle JIK\cong\angle GIH\) (alternate - interior angles). By the SAS (Side - Angle - Side) congruence criterion, \(\triangle IJK\cong\triangle IHG\).

Answer:

1. Given (the problem states \(\triangle GIK\) is equilateral)
2. Vertical - angles theorem
3. Alternate - interior angles theorem
4. Given (from the equilateral triangle \(\triangle GIK\))
5. SAS (Side - Angle - Side) congruence criterion