Question

question 7 of 10
what else would need to be congruent to show that $\triangle jkl \cong \triangle mno$ by aas?

a. $\angle l \cong \angle o$

b. $\overline{lk} \cong \overline{on}$

c. $\overline{lk} \cong \overline{om}$

d. $\angle k \cong \angle n$

Explanation:

Step1: Recall AAS Congruence

AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent.

First, identify the right angles: $\angle J$ and $\angle M$ are right angles, so $\angle J\cong\angle M$. Also, $JK = MN = 3$, so $JK\cong MN$.

Step2: Analyze the given and required for AAS

We already have one pair of angles ($\angle J\cong\angle M$) and we need another pair of angles and a non - included side. Let's look at the triangles $\triangle JKL$ and $\triangle MNO$.

For AAS, we need two angles and a non - included side. We know $\angle J\cong\angle M$ (right angles) and $JK\cong MN$ (sides of length 3). We need another pair of angles. Let's check the options:

• Option A: If $\angle L\cong\angle O$, then we have $\angle J\cong\angle M$, $\angle L\cong\angle O$, and the non - included side $JK\cong MN$ (since in $\triangle JKL$, the side between $\angle J$ and $\angle K$ is $JK$, and in $\triangle MNO$, the side between $\angle M$ and $\angle N$ is $MN$; but with $\angle J\cong\angle M$ and $\angle L\cong\angle O$, the non - included side would be $KL$ and $NO$? Wait, no. Wait, in $\triangle JKL$, angles are $\angle J$ (right), $\angle K$, $\angle L$. In $\triangle MNO$, angles are $\angle M$ (right), $\angle N$, $\angle O$. We know $\angle J=\angle M$ (right angles), $JK = MN = 3$. If we have $\angle L=\angle O$, then by AAS (two angles: $\angle J=\angle M$, $\angle L=\angle O$ and the non - included side $JK = MN$ (since $JK$ is opposite $\angle L$ and $MN$ is opposite $\angle O$)), the triangles would be congruent.

• Option B: $\overline{LK}\cong\overline{ON}$: This would be a side - angle - side or side - side - angle situation, not AAS.

• Option C: $\overline{LK}\cong\overline{OM}$: These are not corresponding sides.

• Option D: $\angle K\cong\angle N$: If $\angle K\cong\angle N$, along with $\angle J\cong\angle M$ and $JK\cong MN$, this would be ASA (Angle - Side - Angle) because the side $JK$ is included between $\angle J$ and $\angle K$, and $MN$ is included between $\angle M$ and $\angle N$.

So, to use AAS, we need two angles and a non - included side. So $\angle L\cong\angle O$ (Option A) gives us two angles ($\angle J\cong\angle M$ and $\angle L\cong\angle O$) and the non - included side $JK\cong MN$.

Answer:

A. $\angle L\cong\angle O$