Question

$\angle fgk \cong \angle hji$, $\overline{gh} \cong \overline{jk}$, and $\angle fkg \cong \angle ihj$. complete the proof that $\triangle hij \cong \triangle kfg$.

statement	reason
2 $\overline{gh} \cong \overline{jk}$	given
3 $\angle fkg \cong \angle ihj$	given
4 $gk = jk + gj$	additive property of length
5 $hj = gh + gj$
6 $gk = gh + gj$	substitution
7 $hj = gk$	transitive property of equality
8 $\triangle hij \cong \triangle kfg$

Explanation:

Step1: Identify missing reason for Step5

The statement $HJ = GH + GJ$ follows from the segment addition postulate, which states that if a point lies on a segment, the total length is the sum of the two smaller segments.

Step2: Identify reason for Step8

The statement $\triangle HIJ \cong \triangle KFG$ uses the Angle-Side-Angle (ASA) Congruence Postulate, as we have two pairs of congruent angles ($\angle FGK \cong \angle HJI$, $\angle FKG \cong \angle IHJ$) and a pair of congruent included sides ($HJ = GK$).

Answer:

1. Reason for Statement 5: Segment Addition Postulate
2. Reason for Statement 8: ASA Congruence Postulate