Question

δgik is equilateral and ∠hgi ≅ ∠ikj. complete the proof that δijk ≅ δihg.
1 δgik is equilateral given
2 ∠hgi ≅ ∠ikj given
3 ∠jik ≅ ∠gih
4 (overline{ik}) ≅ (overline{gi})
5 δijk ≅ δihg

Explanation:

Step1: Identify vertical angles

$\angle JIK \cong \angle GIH$ (Vertical angles are congruent)

Step2: Use equilateral triangle property

$\overline{IK} \cong \overline{GI}$ (Sides of an equilateral triangle are congruent)

Step3: Apply ASA congruence

$\triangle JIK \cong \triangle IHG$ (ASA: $\angle IKJ \cong \angle HGI$, $\overline{IK} \cong \overline{GI}$, $\angle JIK \cong \angle GIH$)

Answer:

1. Vertical angles are congruent
2. Sides of an equilateral triangle are congruent
3. ASA congruence criterion

$\triangle JIK \cong \triangle IHG$