Question

question 1-6
which additional statements are needed to prove $\triangle abc \cong \triangle lkm$ using the angle-angle-side triangle congruence theorem, if $\angle b \cong \angle k$? select all that apply.
$\square$ $\angle b \cong \angle l$
$\square$ $\angle c \cong \angle m$
$\square$ $\overline{ab} \cong \overline{lk}$
$\square$ $\overline{bc} \cong \overline{km}$
$\square$ $\overline{ab} \cong \overline{lm}$

Explanation:

To prove \(\triangle ABC \cong \triangle LKM\) using the Angle - Angle - Side (AAS) Congruence Theorem, we know that AAS requires two angles and a non - included side of one triangle to be congruent to the corresponding two angles and non - included side of the other triangle. We are given that \(\angle B\cong\angle K\).

Step 1: Analyze the option \(\angle B\cong\angle L\)

This option gives us another pair of congruent angles. If we have \(\angle B\cong\angle K\) (given) and \(\angle B\cong\angle L\), this does not directly help us in forming the AAS criteria as we need angles related to the triangles \(\triangle ABC\) and \(\triangle LKM\) in a correct correspondence.

Step 2: Analyze the option \(\angle C\cong\angle M\)

If \(\angle C\cong\angle M\) and we already know that \(\angle B\cong\angle K\), then we have two pairs of congruent angles. Now, for the side, in AAS, the side should be a non - included side. Let's consider the sides. If we have \(\angle B\cong\angle K\), \(\angle C\cong\angle M\), and if we can get a non - included side congruent, we can apply AAS.

Step 3: Analyze the option \(\overline{AB}\cong\overline{LK}\)

This side is not a non - included side with respect to the angles \(\angle B\) and \(\angle C\) (in \(\triangle ABC\)) and \(\angle K\) and \(\angle M\) (in \(\triangle LKM\)). So this does not help in AAS.

Step 4: Analyze the option \(\overline{BC}\cong\overline{KM}\)

This side is the included side between \(\angle B\) and \(\angle C\) in \(\triangle ABC\) and between \(\angle K\) and \(\angle M\) in \(\triangle LKM\). So this is for ASA (Angle - Side - Angle) and not AAS.

Step 5: Analyze the option \(\overline{AB}\cong\overline{LM}\)

If we have \(\angle B\cong\angle K\), \(\angle C\cong\angle M\) (from the second option) and \(\overline{AB}\cong\overline{LM}\), \(\overline{AB}\) is a non - included side with respect to \(\angle B\) and \(\angle C\) in \(\triangle ABC\) and \(\overline{LM}\) is the corresponding non - included side with respect to \(\angle K\) and \(\angle M\) in \(\triangle LKM\). Also, the option \(\angle C\cong\angle M\) gives us the second pair of angles needed for AAS.

The correct additional statements needed are \(\angle C\cong\angle M\) and \(\overline{AB}\cong\overline{LM}\) and \(\angle B\cong\angle L\) is incorrect, \(\overline{BC}\cong\overline{KM}\) is for ASA. Wait, re - evaluating:

The AAS theorem 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.

Given \(\angle B\cong\angle K\) (let's assume the correspondence of vertices \(A ightarrow L\), \(B ightarrow K\), \(C ightarrow M\)).

- For \(\angle C\cong\angle M\): This gives us a second angle congruence (\(\angle B\cong\angle K\) and \(\angle C\cong\angle M\)).
- For \(\overline{AB}\cong\overline{LM}\): If we consider the correspondence \(A ightarrow L\), \(B ightarrow K\), \(C ightarrow M\), \(\overline{AB}\) is a side in \(\triangle ABC\) and \(\overline{LM}\) is the corresponding side in \(\triangle LKM\). With \(\angle B\cong\angle K\) and \(\angle C\cong\angle M\), \(\overline{AB}\cong\overline{LM}\) (non - included side) satisfies AAS.

The option \(\angle B\cong\angle L\) is incorrect as it misrepresents the angle correspondence. The option \(\overline{BC}\cong\overline{KM}\) is an included side (for ASA) not for AAS. The option \(\overline{AB}\cong\overline{LK}\) has incorrect side correspondence.

So the correct additional statements are \(\angle C\cong\angle M\) and \(\overline{AB}\cong\overline{LM}\) (among the given options, the ones that apply are \(\angle C\cong\angle M\) and \(\overline{AB}\cong\overline{LM}\), but looking at the options:

The option \(\angle C\cong\angle M\) (second option) and \(\overline{AB}\cong\overline{LM}\) (fifth option) are the correct ones. Wait, in the given options:

1. \(\angle B\cong\angle L\): Incorrect.
2. \(\angle C\cong\angle M\): Correct (gives a second angle for AAS).
3. \(\overline{AB}\cong\overline{LK}\): Incorrect.
4. \(\overline{BC}\cong\overline{KM}\): Incorrect (included side).
5. \(\overline{AB}\cong\overline{LM}\): Correct (non - included side with respect to the two angles \(\angle B\cong\angle K\) and \(\angle C\cong\angle M\)).

Answer:

The correct additional statements are \(\boldsymbol{\angle C\cong\angle M}\) and \(\boldsymbol{\overline{AB}\cong\overline{LM}}\) (corresponding to the second and fifth options in the list of options: \(\angle C\cong\angle M\) and \(\overline{AB}\cong\overline{LM}\)).