A triangle is cut into several (no less than two) triangles. One of them is isosceles (not equilateral), and the others are equilateral. Find the angles of the original triangle.

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Among the vertices of a scalene triangle, at least one is not a vertex of the original triangle. The sum of the angles of the triangles in the partition that meet at this vertex is $180^{\circ}$ or $360^{\circ}$. Therefore, the angle of the triangle is a multiple of $60^{\circ}$, and since the triangle is not equilateral, this angle is $120^{\circ}$. Then the other two angles of this triangle are $30^{\circ}$, and since they are not multiples of $60^{\circ}$, the corresponding vertices are at the vertices of the original triangle. The angles of the triangle at these vertices can only be $30^{\circ}$, $90^{\circ}$, or $150^{\circ}$, with at least one of the two angles not equal to $30^{\circ}$, and their sum is less than $180^{\circ}$. The only possible option is $30^{\circ}$ and $90^{\circ}$. Such a triangle can be cut in the required manner, for example, by drawing the median from the vertex of the right angle (see figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_dc8ed1d6cdac2bf77445g-27.jpg?height=414&width=914&top_left_y=702&top_left_x=572)

## Answer

$30^{\circ}, 60^{\circ}, 90^{\circ}$.

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