Question

For each diagram, name a pair of overlapping triangles. Tell whether the triangles are congruent by the SSS, SAS, ASA, AAS, or HL method.

Given:$\angle \mathbf{2}\cong \angle \mathbf{3}\mathbf{;}\mathbf{}\angle \mathbf{1}\cong \angle \mathbf{4}$

Short answer

The pair of overlapping triangles that are congruent are$△\mathbf{LMN}$and $△\mathbf{PNM}$.

The triangles$△LMN$ and$△PNM$ are congruent by the $\mathit{A}\mathit{S}\mathit{A}$method.

Step-by-step solution

- Observe the given diagram.

The given diagram is:

- Description of step.

It is being given that$\angle 2\cong \angle 3$and$\angle 1\cong \angle 4$.

As, $\angle 2\cong \angle 3$and $\angle 1\cong \angle 4$,therefore $\angle 2=\angle 3$and $\angle 1=\angle 4$.

Now, it can be obtained that:

$\begin{array}{l}\angle 1+\angle 2=\angle 3+\angle 4\\ \angle LMN=\angle PNM\end{array}$

That implies,$\angle LMN\cong \angle PNM$

In the triangles$△LMN$ and $△PNM$, the side that is common is $MN$.

Therefore,$\overline{MN}\cong \overline{MN}$by using the reflexive property.

As,$\angle 2\cong \angle 3$, thereforerole="math" localid="1648885940559" $\angle PMN\cong \angle LNM$.

In the triangles $△LMN$ and $△PNM$, it can be noticed that$\angle LMN\cong \angle PNM,\overline{MN}\cong \overline{MN}$ and$\angle PMN\cong \angle LNM$.

Therefore, the triangles $△LMN$and $△PNM$are the congruent triangles by using the $ASA$method.

- Write the conclusion.

The pair of overlapping triangles that are congruent are$△LMN$ and $△PNM$.

The triangles$△LMN$ and$△PNM$are congruent by the $ASA$method.