Task B-1.5. An isosceles trapezoid, which has no right angles, is divided by a diagonal into two isosceles triangles. Determine the angles of the trapezoid.
Solution. Sketch

![](https://cdn.mathpix.com/cropped/2024_05_30_cf2c4989ba380bc8684dg-02.jpg?height=558&width=816&top_left_y=1347&top_left_x=627)

We have $\beta+\alpha=180^{\circ}-\alpha$, which means $2 \alpha+\beta=180^{\circ}$.

(1 point)

On the other hand, $S$ is

$$
\begin{aligned}
180^{\circ}-\alpha+\beta+\beta & =180^{\circ} \\
2 \beta & =\alpha
\end{aligned}
$$

Then $2 \cdot 2 \beta+\beta=180^{\circ}$, or $5 \beta=180^{\circ}$, which gives $\beta=36^{\circ}, \alpha=72^{\circ}$.

(1 point)

The angles of the trapezoid are $\alpha=72^{\circ}$ and $\alpha+\beta=108^{\circ}$.