Question

△ghj is equilateral, j is the midpoint of (overline{ik}), and (overline{hi} cong overline{gk}). complete the proof that △gjk ≅ △hij.

statement	reason
2 j is the midpoint of (overline{ik})	given
3 (overline{hi} cong overline{gk})	given
4 (overline{jk} cong overline{ij})	definition of midpoint
5 (overline{gj} cong overline{hj})	definition of equilateral triangle
6

Explanation:

Step1: Identify Triangle Properties

Since \( \triangle GHJ \) is equilateral, \( GJ = HJ \) (by definition of equilateral triangle, all sides are equal). So \( \overline{GJ} \cong \overline{HJ} \) is given by the definition of an equilateral triangle (matches statement 5's reason).

Step2: Midpoint Implication

\( J \) is the midpoint of \( \overline{IK} \), so \( JK = IJ \) (definition of midpoint, statement 4's reason: definition of midpoint).

Step3: Given Congruence

We know \( \overline{HI} \cong \overline{GK} \) (statement 3, given).

Step4: SSS Congruence

Now, in \( \triangle GJK \) and \( \triangle HJI \):

• \( GJ = HJ \) (from equilateral \( \triangle GHJ \))
• \( JK = IJ \) (from midpoint \( J \) of \( IK \))
• \( GK = HI \) (given)

By SSS (Side - Side - Side) congruence criterion, \( \triangle GJK \cong \triangle HJI \). So statement 6 should be \( \triangle GJK \cong \triangle HJI \) and reason is SSS Congruence Postulate.

Answer:

Statement 6: \( \triangle GJK \cong \triangle HJI \)
Reason: SSS (Side - Side - Side) Congruence Postulate