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. \\(\angle b \cong \angle l\\) \\(\angle c \cong \angle m\\) \\(\overline{bl} \cong \overline{lk}\\) \\(\overline{km} \cong \overline{km}\\) \\(\overline{ab} \cong \overline{lm}\\)

Explanation:

Step1: Recall AAS Congruence

AAS (Angle - Angle - Side) congruence 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 two triangles are congruent. We know that $\angle B\cong\angle K$.

Step2: Analyze Each Option

• For $\angle B\cong\angle L$: This doesn't help directly with AAS for $\triangle ABC\cong\triangle LKM$ as we already have $\angle B\cong\angle K$.
• For $\angle C\cong\angle M$: If we have $\angle B\cong\angle K$ (given) and $\angle C\cong\angle M$, then we have two angles. Then, if we have a corresponding non - included side, we can use AAS. Also, if we consider the sides:
• For $\overline{LK}\cong\overline{AB}$: If we have $\angle B\cong\angle K$ and another pair of angles (say $\angle C\cong\angle M$), and $\overline{LK}\cong\overline{AB}$ (a non - included side), this can help.
• For $\overline{KM}\cong\overline{BC}$: If $\angle B\cong\angle K$ and another pair of angles, and $\overline{KM}\cong\overline{BC}$ (a non - included side), this can help.
• For $\overline{AB}\cong\overline{LM}$: This side doesn't seem to be a corresponding non - included side for the AAS of $\triangle ABC$ and $\triangle LKM$ with the given $\angle B\cong\angle K$.

So the additional statements needed are $\angle C\cong\angle M$, $\overline{LK}\cong\overline{AB}$, $\overline{KM}\cong\overline{BC}$. But since the options are likely among the given ones, and based on AAS, if we have $\angle B\cong\angle K$ (given), to apply AAS we need another pair of angles (like $\angle C\cong\angle M$) and a corresponding non - included side. Among the options, $\angle C\cong\angle M$, $\overline{LK}\cong\overline{AB}$, $\overline{KM}\cong\overline{BC}$ are relevant. But from the visible options (assuming the options are $\angle B\cong\angle L$, $\angle C\cong\angle M$, $\overline{LK}\cong\overline{AB}$, $\overline{KM}\cong\overline{BC}$, $\overline{AB}\cong\overline{LM}$), the correct ones are $\angle C\cong\angle M$, $\overline{LK}\cong\overline{AB}$, $\overline{KM}\cong\overline{BC}$. But since the user's image has options, and if we assume the options are as listed (the vertical text), the correct additional statements to prove $\triangle ABC\cong\triangle LKM$ by AAS (given $\angle B\cong\angle K$) are $\angle C\cong\angle M$, $\overline{LK}\cong\overline{AB}$, $\overline{KM}\cong\overline{BC}$. But if we have to pick from the options (the ones with checkboxes), the ones that apply are $\angle C\cong\angle M$, $\overline{LK}\cong\overline{AB}$, $\overline{KM}\cong\overline{BC}$.

Answer:

The additional statements needed are $\boldsymbol{\angle C\cong\angle M}$, $\boldsymbol{\overline{LK}\cong\overline{AB}}$, $\boldsymbol{\overline{KM}\cong\overline{BC}}$ (depending on the exact options, but based on AAS congruence theorem).