Triangles J K L and M N R are shown.

In the diagram, ∠J ≅ ∠M and JL ≅ MR. What additional information is needed to show ΔJKL ≅ △MNR by SAS?

KL ≅ NR

∠L ≅ ∠R

∠K ≅ ∠N

JK ≅ MN

To prove that triangles \( \triangle JKL \) and \( \triangle MNR \) are congruent by the Side-Angle-Side (SAS) criterion, we need two pairs of corresponding sides to be congruent and the included angles (the angles between these pairs of sides) to be congruent.

Given:
- \( \angle J \cong \angle M \)
- \( JL \cong MR \)

Since \( \angle J \) and side \( JL \) are given, we need the following:
1. Another pair of sides, either \( JK \cong MN \) or \( KL \cong NR \), to complete the two pairs of congruent sides.
2. The included angle between \( JL \) and the second side.

Thus, the correct answer is:
**\( JK \cong MN \)** 

This pair of sides, together with the given angle \( \angle J \) (or \( \angle M \) in \( \triangle MNR \)), would satisfy the SAS criterion for triangle congruence.

Would you like further details or have any questions?

Here are five related questions:
1. How does the SAS criterion differ from the ASA criterion in proving triangle congruence?
2. Why is \( \angle J \) considered the "included angle" in the SAS criterion?
3. How could we use the SSS criterion to prove these triangles congruent if all sides were known?
4. What other criteria can be used to prove triangle congruence?
5. Why isn’t \( \angle L \cong \angle R \) sufficient for SAS in this case?

**Tip:** Remember that the SAS criterion requires the angle to be between the two known congruent sides.

Triangles J K L and M N R are shown. In the diagram, ∠J ≅ ∠M and JL ≅ MR. What additional information is needed to show ΔJKL ≅ △MNR by SAS? KL ≅ NR, ∠L ≅ ∠R, ∠K ≅ ∠N, JK ≅ MN

Learn how to prove triangle congruence using the SAS criterion with the given conditions of ∠J ≅ ∠M and JL ≅ MR. The solution involves determining the missing side congruence, JK ≅ MN, to satisfy the SAS theorem. Suitable for Grades 8-10.

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