## Task 2 - 050622

The three lines $g_{1}, g_{2}$, and $g_{3}$ intersect at point $M$. This creates angles with measures $\alpha, \beta, \gamma, \delta, \psi$, and $\omega$ (see figure).

How large are these 6 angle measures if

(1) $\gamma+\delta+\psi+\omega=252^{\circ}$ and

(2) $\alpha$ is three times as large as $\beta$?

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0171.jpg?height=411&width=516&top_left_y=1314&top_left_x=1244)
It is given: $\alpha+\beta+\gamma+\delta+\psi+\omega=360^{\circ}$

and due to (1) $\alpha+\beta=108^{\circ}$, so due to (2) $4 \beta=108^{\circ}$, i.e., $\beta=27^{\circ}$ and therefore due to (2) $\alpha=81^{\circ}$.

Furthermore, there are vertical angles, i.e., $\delta=\alpha=81^{\circ}$ and $\psi=\beta=27^{\circ}$. According to the adjacent angle relationship, $(\alpha+\beta)+\gamma=180^{\circ}$ and thus $\gamma=\omega=72^{\circ}$.