In the quadrilateral $K L M N$, we know the marked angles and that $|K N|=|L M|$. What is the measure of angle $K N M$?

![](https://cdn.mathpix.com/cropped/2024_04_17_e23bba49e38f63edc998g-3.jpg?height=377&width=434&top_left_y=451&top_left_x=845)

(L. Hozová)
Since the sum of the interior angles in any triangle is $180^{\circ}$, the measure of angle $L K M$ is $180^{\circ}-75^{\circ}-30^{\circ}=75^{\circ}$. Therefore, triangle $K L M$ is isosceles, i.e., $|L M|=|K M|$. According to the problem, $|L M|=|K N|$, thus $|K M|=|K N|$ and triangle $K M N$ is also isosceles. The measure of angle $K N M$ is therefore

$$
\left(180^{\circ}-50^{\circ}\right): 2=65^{\circ} .
$$