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 \(\triangle JKL \cong \triangle MNR\) by SAS (Side - Angle - Side) congruence criterion, we need two sides and the included angle of one triangle to be congruent to the corresponding two sides and the included angle of the other triangle.

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 satisfy the SAS criterion.

Let's analyze the other options:

• Option \(\angle J\cong\angle M\): \(\angle J\) and \(\angle M\) are not the included angles between the given congruent sides, so this does not satisfy SAS.
• Option \(\angle K\cong\angle N\): \(\angle K\) and \(\angle N\) are not the included angles between the given congruent sides, so this does not satisfy SAS.
• Option \(\angle R\cong\angle K\): These angles are not the included angles between the given congruent sides, so this does not satisfy SAS.

Answer:

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