Question

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

Explanation:

Step1: Recall segment - addition postulate

By the segment - addition postulate, $GI=GH + HI$ and $JL=JK + KL$.

Step2: Use the given congruences

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

Step3: Substitute and simplify

Substitute $GI=GH + HI$ and $JL=JK + KL$ into $GI = JL$. We get $GH + HI=JK + KL$. Then substitute $GH = KL$ into $GH + HI=JK + KL$. After substitution, $KL+HI=JK + KL$. Subtract $KL$ from both sides of the equation $KL + HI=JK + KL$ (subtraction property of equality). We obtain $HI = JK$.

Step4: Use the definition of congruent segments

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

Answer:

Statements	Reasons
2. $GI = JL$, $GH = KL$	Definition of congruent segments
3. $GI=GH + HI$, $JL=JK + KL$	Segment - addition postulate
4. $GH + HI=JK + KL$	Substitution property of equality (from 2 and 3)
5. $KL+HI=JK + KL$	Substitution property of equality (substitute $GH = KL$ into 4)
6. $HI = JK$	Subtraction property of equality
7. $\overline{HI}\cong\overline{JK}$	Definition of congruent segments