Question

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Explanation:

Step1: Recall Similar Triangles Criterion

We use the AA (Angle - Angle) similarity criterion. In right - angled triangles, if two angles are equal, the triangles are similar.

Step2: Analyze Angles in $\triangle LMN$ and $\triangle LMK$

• $\triangle LMN$ and $\triangle LMK$: $\angle L$ is common to both triangles, and $\angle LNM=\angle LMK = 90^{\circ}$. By AA similarity, $\triangle LMN\sim\triangle LMK$.

Step3: Analyze Angles in $\triangle LMN$ and $\triangle MKN$

• $\angle MNL=\angle MKN = 90^{\circ}$, and $\angle LMN=\angle MK N$ (because $\angle LMN+\angle NMK = 90^{\circ}$ and $\angle NMK+\angle MK N=90^{\circ}$, so $\angle LMN=\angle MK N$). Also, $\angle L$ is related, but more directly, using AA similarity: $\angle MNL=\angle MKN = 90^{\circ}$ and $\angle L=\angle L$ (for $\triangle LMN$ and $\triangle LMK$) or $\angle LMN=\angle MK N$ and $\angle MNL=\angle MKN$ (for $\triangle LMN$ and $\triangle MKN$). Wait, actually, in the diagram, $\triangle LMN$, $\triangle LMK$, and $\triangle MKN$ are all right - angled.
• For $\triangle LMN$ and $\triangle MKN$: $\angle MNL=\angle MKN = 90^{\circ}$, and $\angle LMN=\angle MK N$ (as shown by the right - angle and angle - sum property). So by AA similarity, $\triangle LMN\sim\triangle MKN$. Also, $\triangle LMN\sim\triangle LMK$ (since $\angle L$ is common and both are right - angled).

Answer:

$\triangle LMN\sim\triangle LMK$; $\triangle LMN\sim\triangle MKN$ (or $\triangle LMN\sim\triangle LMK$ and $\triangle LMN\sim\triangle MKN$ depending on the diagram's exact angle relationships. The two similar triangles to $\triangle LMN$ are $\triangle LMK$ and $\triangle MKN$)