Question

what statement can be given for line 4? △hfg≅△ihf △fhg≅△ihf △gfh≅△ifh fg≅if fh≅fh

Explanation:

Step1: Analyze triangle congruence notation

Triangle congruence notation requires that the vertices of the congruent triangles are listed in corresponding order.

Step2: Check given options

• For \(\triangle HFG\cong\triangle IHF\), the vertex order does not match the corresponding parts.
• For \(\triangle FHG\cong\triangle IHF\), the vertex order does not match the corresponding parts.
• For \(\triangle GFH\cong\triangle IFH\), the vertices are in corresponding order. When proving triangle congruence (assuming previous steps like \(GF = IF\) (from step 2, \(GF\cong IF\)), \(FH = FH\) (common side, step 3, \(FH\cong FH\)) and \(\angle GFH=\angle IFH\) (from step 1, \(\angle GFH,\angle IFH\) given), by \(SAS\) (Side - Angle - Side) congruence criterion, \(\triangle GFH\cong\triangle IFH\)
• \(FG\cong IF\) is a side - congruence, not a triangle - congruence statement for line 4 (which is supposed to be a triangle - congruence statement as per the proof structure)
• \(FH\cong FH\) is a side - congruence (already in step 3)

Answer:

\(\triangle GFH\cong\triangle IFH\)