To prove that line segments \( \overline{IJ} \) and \( \overline{FH} \) are parallel based on the given triangles, we can use the fact that corresponding angles are congruent when two angles are formed by a transversal intersecting parallel lines.
Since Sage has already proven that triangles \( \triangle IGJ \) and \( \triangle FGH \) are similar, we know that their corresponding angles are congruent.
Thus, the missing statement and reason in Sage's proof are:
Missing Statement: \( \angle GJI \cong \angle GHF \)
Missing Reason: "Corresponding angles of similar triangles are congruent."
Hence the correct response is:
The missing statement is \( \angle GJI \cong \angle GHF \) and the missing reason is "corresponding angles of similar triangles are congruent."