Question

△gik is equilateral. complete the proof that △ghi ≅ △kji.

1 △gik is equilateral given
2 ∠h ≅ ∠j given
3 ∠jik ≅ ∠gih
4 overline{ik} ≅ overline{gi}
5 △ghi ≅ △kji

Explanation:

Step1: Vertical angles are congruent

$\angle JIK\cong\angle GIH$ (Vertical angles theorem)

Step2: Sides of equilateral triangle

Since $\triangle GIK$ is equilateral, all its sides are equal. So, $\overline{IK}\cong\overline{GI}$ (Definition of equilateral triangle)

Step3: AAS congruence criterion

In $\triangle GHI$ and $\triangle KJI$, we have $\angle H\cong\angle J$ (given), $\angle GIH\cong\angle JIK$ (vertical angles), and $\overline{GI}\cong\overline{IK}$ (sides of equilateral triangle). So, $\triangle GHI\cong\triangle KJI$ (AAS - Angle - Angle - Side congruence criterion)

Answer:

1. Vertical angles theorem; 4. Definition of equilateral triangle; 5. AAS (Angle - Angle - Side)