Question

△gik is equilateral. complete the proof that △ijk ≅ △ihg.

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

Explanation:

Step1: Reason for Statement 1

Given that \(\triangle GIK\) is equilateral, this is the given information for the proof.

Step2: Reason for Statement 2

\(\angle IKJ \cong \angle HGI\) – these angles are marked as congruent in the diagram (the red arcs), so the reason is "Given (marked congruent angles)".

Step3: Reason for Statement 3

\(\angle JIK \cong \angle GIH\) – these are vertical angles, and vertical angles are congruent. So the reason is "Vertical Angles Theorem".

Step4: Reason for Statement 4

Since \(\triangle GIK\) is equilateral, all its sides are congruent, so \(IK = GI\) (or \(\overline{IK} \cong \overline{GI}\)). The reason is "Sides of an equilateral triangle are congruent".

Step5: Reason for Statement 5

To prove \(\triangle IJK \cong \triangle IHG\), we have two angles and a included side? Wait, let's check the angles and sides. We have \(\angle IKJ \cong \angle HGI\) (Statement 2), \(\angle JIK \cong \angle GIH\) (Statement 3), and \(\overline{IK} \cong \overline{GI}\) (Statement 4). So by the Angle - Angle - Side (AAS) Congruence Theorem, \(\triangle IJK \cong \triangle IHG\). The reason is "AAS (Angle - Angle - Side) Congruence Theorem".

Answer:

1. Reason: Given
2. Reason: Given (marked congruent angles)
3. Reason: Vertical Angles Theorem
4. Reason: Sides of an equilateral triangle are congruent
5. Reason: AAS (Angle - Angle - Side) Congruence Theorem