A trapezoid $A B C D(A B \| C D)$ with equal non-parallel sides is given, where $|A B|>|C D|$. Through point $A$, two lines can be drawn to divide the trapezoid into three isosceles triangles. Determine the sizes of the angles of trapezoid $A B C D$.
It is evident that one of the bisecting lines must pass through point $C$, and the second bisecting line will intersect either the base $CD$ or the side $BC$ at point $E$.

a) Suppose the second bisecting line intersects the side $BC$:

![](https://cdn.mathpix.com/cropped/2024_04_17_c1ee7b21a76ccad0da1cg-1.jpg?height=328&width=783&top_left_y=1492&top_left_x=1820)

Triangle $ADC$ with an obtuse angle at vertex $D$ can only be isosceles if $|AD| = |CD|$. In triangle $AEB$, the angle at vertex $A$ is smaller than the angle at vertex $B$, so $|AE| > |EB|$. At the same time, $|AB| > |EB|$, because $|EB| < |CD|$, and it is clear that angle $ADE$ is obtuse. Therefore, $|AD| = |DE|$. Similarly, angle $AEC$ must also be obtuse, so $|AE| = |EC|$. In triangle $ABC$, the angle at vertex $A$ is smaller than the angle at vertex $B$, so $|AC| > |BC|$. At the same time, $|BC| < |AB|$, because $|BC| = |AD| = |DE| < |DC| < |AB|$. For $ABC$ to be isosceles, it must therefore be that $|AB| = |AC|$. Let $\alpha$ be the angle $ABC$, and we will sequentially express the internal angles of the isosceles triangles being considered:

![](https://cdn.mathpix.com/cropped/2024_04_17_c1ee7b21a76ccad0da1cg-2.jpg?height=328&width=784&top_left_y=1253&top_left_x=354)

Since the trapezoid is isosceles, it follows that $|\angle DAB| = \alpha$, i.e.,

$$
\left(360^{\circ} - 4\alpha\right) + \left(180^{\circ} - 2\alpha\right) + \left(180^{\circ} - 2\alpha\right) = \alpha.
$$

From this, we get that

$$
\alpha = 80^{\circ}.
$$

The internal angles of the trapezoid are therefore $80^{\circ}, 80^{\circ}, 100^{\circ}, 100^{\circ}$

[2 points for excluding the possibility that the second bisecting line intersects the side $BC$ at 2 points, 2 points for setting up the equation, 2 points for the correct solution of the equation]