Question

1. given: $overline{gi}congoverline{jl}$, $overline{gh}congoverline{kl}$ prove: $overline{hi}congoverline{jk}$ statements reasons $overline{gi}congoverline{jl},overline{gh}congoverline{kl}$ given $gi = jl,gh = kl$ $gi=gh + hi,jl=jk + kl$ $gh + hi=jk + kl$ $hi = jk$ $overline{hi}congoverline{jk}$ © gina wilson (all things algebra*, llc). 2014

Explanation:

Step1: Define congruent segments

If $\overline{GI}\cong\overline{JL}$ and $\overline{GH}\cong\overline{KL}$, then $GI = JL$ and $GH=KL$ (by the definition of congruent segments, congruent segments have equal lengths).

Step2: Segment - addition postulate

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

Step3: Substitute equal lengths

Since $GI = JL$, we can substitute $GI$ with $GH + HI$ and $JL$ with $JK + KL$, getting $GH + HI=JK + KL$.

Step4: Subtract equal quantities

Given $GH = KL$, we subtract $GH$ (or $KL$) from both sides of the equation $GH + HI=JK + KL$. So, $HI=JK$.

Step5: Define congruent segments again

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

Answer:

The reasons for each statement in order are: Given; Definition of congruent segments; Segment - addition postulate; Substitution property of equality; Subtraction property of equality; Definition of congruent segments.