Question

in the diagram, \\( \angle j \cong \angle m \\) and \\( \overline{jl} \cong \overline{mr} \\). what additional information is needed to show \\( \triangle jkl \cong \triangle mnr \\) by sas?
\\( \bigcirc \overline{kl} \cong \overline{nr} \\)
\\( \bigcirc \angle l \cong \angle r \\)
\\( \bigcirc \angle k \cong \angle n \\)
\\( \bigcirc \overline{jk} \cong \overline{mn} \\)

Explanation:

Step1: Recall the SAS (Side - Angle - Side) congruence criterion

For two triangles $\triangle ABC$ and $\triangle DEF$, $\triangle ABC\cong\triangle DEF$ by SAS if $AB = DE$, $\angle A=\angle D$, and $AC = DF$.

Step2: Identify the given information in $\triangle JKL$ and $\triangle MNR$

We are given that $\angle J\cong\angle M$ and $\overline{JL}\cong\overline{MR}$.

Step3: Determine the missing side for SAS

In the SAS congruence, the angle is between two sides. For $\triangle JKL$ and $\triangle MNR$, with $\angle J$ (in $\triangle JKL$) and $\angle M$ (in $\triangle MNR$) as the included angles, we already have one pair of sides ($\overline{JL}\cong\overline{MR}$). The other pair of sides adjacent to the given angles ($\angle J$ and $\angle M$) are $\overline{JK}$ and $\overline{MN}$.

So, if $\overline{JK}\cong\overline{MN}$, then by the SAS (Side - Angle - Side) congruence criterion, $\triangle JKL\cong\triangle MNR$.

Answer:

$\overline{JK}\cong\overline{MN}$