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 \(\overline{JL}\cong\overline{MR}\). Let's analyze the sides and the angles:

• 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, for the SAS criterion, we need the included angle between the two given congruent sides to be congruent. That is, we need \(\angle L\cong\angle R\).

Let's check 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 related to the included angles of the given congruent sides.

Answer:

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