Question

complete the proof that $\triangle fgkcong\triangle ijh$.

statement	reason
2	$overline{hi}congoverline{fk}$	given
3	$overline{jk}congoverline{gh}$	given
4	$gk = jk+gj$
5	$hj = gh + gj$
6	$gk = gh+gj$	substitution
7	$hj = gk$
8	$\triangle fgkcong\triangle ijh$	sss

Explanation:

Step1: Segment - addition postulate

By the segment - addition postulate, if a point $J$ lies on line segment $\overline{GK}$, then $GK = JK+GJ$.

Step2: Segment - addition postulate

By the segment - addition postulate, if a point $G$ lies on line segment $\overline{HJ}$, then $HJ = GH + GJ$.

Step3: Substitution

Since $JK\cong GH$ (given), we can substitute $JK$ with $GH$ in the equation $GK = JK+GJ$ to get $GK=GH + GJ$.

Step4: Transitive property

Since $HJ = GH + GJ$ and $GK=GH + GJ$, by the transitive property of equality, $HJ = GK$.

Step5: SSS congruence

We have $\overline{FG}\cong\overline{IJ}$ (given), $\overline{HI}\cong\overline{FK}$ (given), and $\overline{GK}\cong\overline{HJ}$ (proven above). So, by the Side - Side - Side (SSS) congruence criterion, $\triangle FGK\cong\triangle IJH$.

Answer:

1. Segment - addition postulate
2. Segment - addition postulate
3. Transitive property of equality