Write a detailed plan for proof.

Given: $\overline{\mathrm{FL}}\cong \overline{\mathrm{AK}};\overline{\mathrm{SF}}\cong \overline{\mathrm{SK}};\mathbf{M}$is the midpoint of $\overline{\mathrm{SF}};\mathbf{N}$is the midpoint of $\overline{\mathrm{SK}}$.

Prove:$\overline{\mathrm{AM}}\cong \overline{\mathrm{LN}}$

The given diagram is:

It is being given that $\overline{FL}\cong \overline{AK};\overline{SF}\cong \overline{SK};M$is the midpoint of$\overline{SF};N$is the midpoint of $\overline{SK}$.

As, $\overline{FL}\cong \overline{AK}$, therefore, $\overline{FL}=\overline{AK}$.

As,$\overline{FL}=\overline{AK}$ therefore it can be noticed that:

$\begin{array}{c}\overline{FL}=\overline{AK}\\ \overline{FL}+\overline{LA}=\overline{AK}+\overline{LA}\\ \overline{FA}=\overline{KL}\end{array}$

Therefore, $\overline{FA}\cong \overline{KL}$.

As, $\overline{SF}\cong \overline{SK}$, therefore $\overline{SF}=\overline{SK}$.

As, $\overline{SF}=\overline{SK}$, therefore the angles$\angle SKF$ and$\angle SFK$that are opposite to the equal sides$SF$and$SK$are also equal.

Therefore, $\angle SKF\cong \angle SFK$.

That implies,$\angle NKF\cong \angle MFK$

As, $M$is the midpoint of$\overline{SF}$.

Therefore, by using the definition of midpoint it can be said that $\overline{SM}=\overline{MF}$.

By using the segment addition postulate it can be noticed that:

$\overline{SF}=\overline{SM}+\overline{MF}$

As, $\overline{SM}=\overline{MF}$, therefore it can be noticed that:

$$\begin{gathered} \overline{SF}=\overline{SM}+\overline{MF} \\ =\overline{MF}+\overline{MF} \\ =2\overline{MF} \end{gathered}$$

As, $N$is the midpoint of$\overline{SK}$.

Therefore, by using the definition of midpoint it can be said that $\overline{SN}=\overline{NK}$.

By using the segment addition postulate it can be noticed that:

$\overline{SK}=\overline{SN}+\overline{NK}$

As, $\overline{SN}=\overline{NK}$, therefore it can be noticed that:

$$\begin{gathered} \overline{SK}=\overline{SN}+\overline{NK} \\ =\overline{NK}+\overline{NK} \\ =2\overline{NK} \end{gathered}$$

As, $\overline{SF}=\overline{SK}$, therefore it can be noticed that:

$\begin{array}{c}\overline{SF}=\overline{SK}\\ 2\overline{MF}=2\overline{NK}\\ \overline{MF}=\overline{NK}\end{array}$

Therefore, $\overline{MF}\cong \overline{NK}$.

Therefore, in the triangles $△AMF$and $△LNK$, it can be noticed that $\overline{MF}\cong \overline{NK},\angle NKF\cong \angle MFK$and $\overline{FA}\cong \overline{KL}$.

Therefore, the triangles$△AMF$ and$△LNK$ are the congruent angles by using the SAS postulate.

The triangles$△AMF$ and$△LNK$are the congruent triangles.

Therefore, by using the corresponding parts of congruent triangles it can be said that $\overline{AM}\cong \overline{LN}$.