Question

what additional information would be needed to prove that the triangles are congruent using the asa congruence theorem? (overline{on}congoverline{mn}) (angle loncongangle lmn) (angle lnocongangle lnm) (overline{ln}congoverline{nm})

Explanation:

Step1: Recall ASA congruence theorem

ASA (Angle - Side - Angle) states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the two triangles are congruent.

Step2: Analyze given figure

We can see that side $\overline{LN}$ is common to both $\triangle LON$ and $\triangle LMN$. There is a marked angle at $L$ which is common to both triangles. We need another pair of angles that share side $\overline{LN}$ as the included side.

Step3: Determine required angle

The angles $\angle LNO$ and $\angle LNM$ share side $\overline{LN}$ as the included side. If $\angle LNO\cong\angle LNM$, along with the common side $\overline{LN}$ and the common angle at $L$, we can use ASA to prove $\triangle LON\cong\triangle LMN$.

Answer:

$\angle LNO\cong\angle LNM$