Question

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

Explanation:

To prove two triangles congruent by SAS (Side - Angle - Side), the angle must be the included angle between the two given sides. In $\triangle JKL$ and $\triangle MNR$, we know that $\overline{KL}\cong\overline{NR}$ and $\overline{JL}\cong\overline{MR}$. The included angle between $\overline{KL}$ and $\overline{JL}$ in $\triangle JKL$ is $\angle L$, and the included angle between $\overline{NR}$ and $\overline{MR}$ in $\triangle MNR$ is $\angle R$. So we need $\angle L\cong\angle R$ to apply the SAS congruence criterion.

• For $\angle J\cong\angle M$: $\angle J$ is not the included angle between the given sides $\overline{KL}$ and $\overline{JL}$, so this does not satisfy SAS.
• For $\angle K\cong\angle N$: $\angle K$ is not the included angle between the given sides $\overline{KL}$ and $\overline{JL}$, so this does not satisfy SAS.
• For $\angle R\cong\angle K$: These angles are not the included angles between the respective sides, so this does not satisfy SAS.

Answer:

$\boldsymbol{\angle L\cong\angle R}$ (the option with the text "$\angle L\cong\angle R$")