In a right triangle $A B C$ with a right angle at vertex $B$, the median $B M$ is drawn. The incircle of triangle $A B M$ touches sides $A B$ and $A M$ at points $K$ and $L$. It is known that lines $K L$ and $B M$ are parallel. Find the angle $C$.

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$A K=A L$, as tangents drawn to a circle from one point. By Thales' theorem, $A M=A B$. On the other hand, $B M=A M$ as the median of a right triangle drawn to the hypotenuse. Therefore, triangle $A B M$ is equilateral, and $\angle A=60^{\circ}$.

## Answer

$30^{\circ}$.