Question

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

Explanation:

Step1: State given information

Given: $\overline{GI}\cong\overline{JL}$, $\overline{GH}\cong\overline{KL}$, $\overline{HI}\cong\overline{JK}$

Step2: Recall SSS (Side - Side - Side) congruence criterion

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. Consider $\triangle GHI$ and $\triangle JKL$.
In $\triangle GHI$ and $\triangle JKL$, we have:

• Side $\overline{GI}$ in $\triangle GHI$ is congruent to side $\overline{JL}$ in $\triangle JKL$ (given).
• Side $\overline{GH}$ in $\triangle GHI$ is congruent to side $\overline{KL}$ in $\triangle JKL$ (given).
• Side $\overline{HI}$ in $\triangle GHI$ is congruent to side $\overline{JK}$ in $\triangle JKL$ (given).

By SSS congruence criterion, $\triangle GHI\cong\triangle JKL$.
Since corresponding parts of congruent triangles are congruent (CPCTC), we can conclude that $\overline{HI}\cong\overline{JK}$ which is what we needed to prove.

Answer:

The proof is completed by using the SSS congruence criterion for $\triangle GHI$ and $\triangle JKL$ and then applying CPCTC.