Question

△fhi ≅ △gih. complete the proof that △fgh ≅ △gfi.
1 △fhi ≅ △gih given
2 overline{fi} ≅ overline{gh}
3 overline{fh} ≅ overline{gi}
4 overline{fg} ≅ overline{fg} reflexive property of congruence
5 △fgh ≅ △gfi

Explanation:

Step1: Corresponding parts of congruent triangles

Since \(\triangle FHI\cong\triangle GIH\) (given), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, \(\overline{FI}\cong\overline{GH}\) and \(\overline{FH}\cong\overline{GI}\).

Step2: SSS congruence criterion

We have \(\overline{FG}\cong\overline{FG}\) (Reflexive Property of Congruence), \(\overline{FH}\cong\overline{GI}\), and \(\overline{FI}\cong\overline{GH}\). For \(\triangle FGH\) and \(\triangle GFI\), if we consider the side - side - side (SSS) congruence criterion. In \(\triangle FGH\) and \(\triangle GFI\), we have:

• \(FG = FG\) (common side)
• \(FH=GI\) (from CPCTC of \(\triangle FHI\cong\triangle GIH\))
• \(GH = FI\) (from CPCTC of \(\triangle FHI\cong\triangle GIH\))

Answer:

1. Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
2. Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
3. Reason: Side - Side - Side (SSS) Congruence Criterion