Question

statement reason
1 ∠gih ≅ ∠ejf given
2 ij ≅ fg given
3 ∠hgi ≅ ∠efj given
4 fj = ij + fi
5 gi = fg + fi
6 fj = fg + fi substitution
7 gi = fj transitive property of equality
8 △efj ≅ △hgi asa

Explanation:

Step1: Segment addition postulate

Segments on a line can be added. So, \( FJ = IJ+FI \) (segment addition postulate).

Step2: Segment addition postulate

Similarly, \( GI = FG + FI \) (segment addition postulate).

Step3: Substitution

Since \( IJ\cong FG \) (given, so \( IJ = FG \)), substitute \( IJ\) with \( FG\) in \( FJ = IJ+FI \) to get \( FJ = FG + FI \).

Step4: Transitive property

Since \( GI = FG + FI \) and \( FJ = FG + FI \), by transitive property \( GI = FJ \).

Step5: ASA congruence

We have \( \angle GIH\cong\angle EJF \) (given), \( \angle HGI\cong\angle EFJ \) (given), and \( GI = FJ \) (from step 4). So, \( \triangle EFJ\cong\triangle HGI \) by ASA (Angle - Side - Angle) congruence criterion.

Answer:

The reasons for statements 4 and 5 are "Segment Addition Postulate".