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?
options:
$angle j cong angle m$
$angle k cong angle n$
$angle r cong angle k$
$angle l cong angle r$

Explanation:

To show \(\triangle JKL \cong \triangle MNR\) by SAS (Side - Angle - Side) congruence criterion, we know that two sides are already given as congruent: \(\overline{KL}\cong\overline{NR}\) and \(\overline{JL}\cong\overline{MR}\). The SAS criterion requires that the included angle between these two sides is congruent.

For \(\triangle JKL\), the sides \(\overline{KL}\) and \(\overline{JL}\) meet at angle \(L\). For \(\triangle MNR\), the sides \(\overline{NR}\) and \(\overline{MR}\) meet at 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.
• Option \(\angle K\cong\angle N\): \(\angle K\) and \(\angle N\) are not the included angles between the given congruent sides.
• Option \(\angle R\cong\angle K\): These angles are not the included angles between the given congruent sides.

Answer:

\(\angle L\cong\angle R\)