Question

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

Explanation:

To prove two triangles congruent by SAS (Side - Angle - Side), 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 $\angle J\cong\angle M$ and $\overline{JL}\cong\overline{MR}$. For SAS, the angle $\angle J$ (or $\angle M$) should be the included angle between two sides. The sides forming $\angle J$ in $\triangle JKL$ are $\overline{JL}$ and $\overline{JK}$, and the sides forming $\angle M$ in $\triangle MNR$ are $\overline{MR}$ and $\overline{MN}$. So, if we have $\overline{JK}\cong\overline{MN}$, then we have two sides ($\overline{JL}\cong\overline{MR}$ and $\overline{JK}\cong\overline{MN}$) and the included angle ($\angle J\cong\angle M$) congruent, which satisfies SAS.

• Option 1: $\overline{KL}\cong\overline{NR}$: These are not the sides forming the known congruent angle, so this does not satisfy SAS.
• Option 2: $\angle L\cong\angle R$: This is an angle - angle - side or angle - side - angle - type information, not SAS.
• Option 3: $\angle K\cong\angle N$: This is also not related to the SAS criterion as it is not the included angle between the known congruent side and the side we need.
• Option 4: $\overline{JK}\cong\overline{MN}$: This provides the second side needed for the SAS criterion with the known angle $\angle J\cong\angle M$ and side $\overline{JL}\cong\overline{MR}$.

Answer:

D. $\overline{JK}\cong\overline{MN}$