Question

1. given: $overline{gi}congoverline{jl}$, $overline{gh}congoverline{kl}$ prove: $overline{hi}congoverline{jk}$

Explanation:

Step1: Recall segment - addition postulate

We know that $GI = GH+HI$ and $JL=JK + KL$ because of the segment - addition postulate.

Step2: Use the given congruences

Since $\overline{GI}\cong\overline{JL}$, then $GI = JL$ (by the definition of congruent segments). Also, since $\overline{GH}\cong\overline{KL}$, then $GH = KL$ (by the definition of congruent segments).

Step3: Substitute the segment - addition expressions

Substitute $GI = GH+HI$ and $JL=JK + KL$ into $GI = JL$. We get $GH + HI=JK + KL$.

Step4: Subtract equal segments

Subtract $GH$ (which is equal to $KL$) from both sides of the equation $GH + HI=JK + KL$. So, $HI=JK$.

Step5: Use the definition of congruent segments

Since $HI = JK$, then $\overline{HI}\cong\overline{JK}$ (by the definition of congruent segments).

Answer:

Statements	Reasons
$GI = JL,GH = KL$	Definition of congruent segments
$GI=GH + HI,JL=JK + KL$	Segment - addition postulate
$GH + HI=JK + KL$	Substitution property
$HI = JK$	Subtraction property of equality
$\overline{HI}\cong\overline{JK}$	Definition of congruent segments