Problem 8. Given an isosceles triangle $K L M(K L=L M)$ with the angle at the vertex equal to $114^{\circ}$. Point $O$ is located inside triangle $K L M$ such that $\angle O M K=30^{\circ}$, and $\angle O K M=27^{\circ}$. Find the measure of angle $\angle L O M$.
Answer: $150^{\circ}$.

Solution. Let $L H$ be the height/median/bisector of the triangle. Let $S$ be the intersection of ray $M O$ and segment $L H$. Note that $K S=S M$. For example, since in triangle $K S M$ the median SH coincides with the height.

![](https://cdn.mathpix.com/cropped/2024_05_06_5f8906e33c6403142659g-05.jpg?height=562&width=1537&top_left_y=1084&top_left_x=268)

Let's calculate the angles: