Question

complete the proof that △efj≅△hgi.
statement reason
1 ij≅fg given
2 ej≅hi given
3 gh≅ef given
4 fj = ij + fi given
5 gi = fg + fi additive property of length
6 fj = fg + fi
7 gi = fj
8 △efj≅△hgi sss

Explanation:

Step1: Substitute using given equality

Since $\overline{IJ}\cong\overline{FG}$, in the equation $FJ = IJ+FI$ (from statement 4), we can substitute $IJ$ with $FG$. So, $FJ=FG + FI$ by the Substitution Property.

Step2: Transitive property application

We know from statement 5 that $GI = FG+FI$ and from step 1 that $FJ = FG + FI$. By the Transitive Property of Equality, if $a=b$ and $c = b$, then $a = c$. So, $GI=FJ$.

Step3: Prove triangle - congruence

We have $\overline{IJ}\cong\overline{FG}$, $\overline{EJ}\cong\overline{HI}$, $\overline{GH}\cong\overline{EF}$ (given) and $GI = FJ$ (from step 2). By the Side - Side - Side (SSS) Congruence Postulate, which states that if the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent, $\triangle EFJ\cong\triangle HGI$.

Answer:

1. Substitution Property
2. Transitive Property of Equality
3. SSS Congruence Postulate