Question

j is the midpoint of \\( \overline{hi} \\). complete the proof that \\( \angle gji \cong \angle gjh \\).

Explanation:

1. The first statement is given in the problem.
2. The marks on \( \overline{GH} \) and \( \overline{GI} \) indicate they are congruent (given by the diagram).
3. A midpoint divides a segment into two congruent parts, so \( \overline{HJ} \cong \overline{IJ} \).
4. \( \overline{GJ} \) is a common side to both triangles, hence congruent to itself.
5. With three pairs of congruent sides (\( \overline{GH} \cong \overline{GI} \), \( \overline{HJ} \cong \overline{IJ} \), \( \overline{GJ} \cong \overline{GJ} \)), the triangles are congruent by SSS.
6. Corresponding parts of congruent triangles are congruent (CPCTC), so the angles are congruent.

Answer:

Statement	Reason
2. \( \overline{GH} \cong \overline{GI} \)	Given (diagram marks)
3. \( \overline{HJ} \cong \overline{IJ} \)	Definition of midpoint
4. \( \overline{GJ} \cong \overline{GJ} \)	Reflexive property of congruence
5. \( \triangle GHJ \cong \triangle GIJ \)	SSS congruence postulate
6. \( \angle GJI \cong \angle GJH \)	CPCTC (Corresponding Parts of Congruent Triangles are Congruent)