## Task 4 - 150924

When solving the task of constructing a triangle $A B C$ from $A B=c, B C=a$ and $\angle B A C=\alpha$, two non-congruent triangles $A B C_{1}$ and $A B C_{2}$ have arisen that satisfy the conditions.

Determine the size of the angle $\angle A C_{1} B$ under these assumptions, given that it is four times as large as the angle $\angle A C_{2} B$!
It is the triangle $B C_{1} C_{2}$ is isosceles with base $C_{1} C_{2}$. Therefore,

$$
\angle B C_{1} C_{2}=\angle C_{1} C_{2} B=\angle A C_{2} B
$$

Since the angles $\angle A C_{1} B$ and $\angle B C_{1} C_{2}$ are supplementary angles, they add up to $180^{\circ}$. Together with $\angle A C_{1} B=4 \cdot \angle A C_{2} B$, it follows that $5 \angle A C_{2} B=180^{\circ}$, thus $\angle A C_{2} B=36^{\circ}$ and finally $\angle A C_{1} B=$ $4 \cdot 36^{\circ}=144^{\circ}$.