8. Variant 1. In triangle $A B C$, the bisector $A L$ and the median $B M$ are drawn. It turns out that $A B=2 B L$. What is the measure of angle $B C A$, if $\angle L M A=127^{\circ}$?
Answer: $74^{\circ}$.

Solution.

![](https://cdn.mathpix.com/cropped/2024_05_06_5ba71d7945ae511dbda6g-5.jpg?height=453&width=753&top_left_y=1752&top_left_x=686)

By the property of the angle bisector, $\frac{A C}{C L}=\frac{A B}{B L}=\frac{2}{1}$, so $A C=2 C L$. Since $M$ is the midpoint of $A C$, then $A M=$ $M C=C L$. In the isosceles triangle $C M L$, the angle $C$ is $180^{\circ}-2 \angle C M L=180^{\circ}-2\left(180^{\circ}-\right.$ $\angle L M A)=2 \angle L M A-180^{\circ}=74^{\circ}$.