# Task 8.5

For a triangle, one of whose angles is equal to $120^{\circ}$, it is known that it can be cut into two isosceles triangles. What can the other two angles of the original triangle be?

## Number of points 7
Answer:

$40^{\circ}$ and $20^{\circ}$ or $45^{\circ}$ and $15^{\circ}$.

## Solution

Let in triangle $ABC$, $\angle B=120^{\circ}$. The cut specified in the problem must pass through one of the vertices of the triangle (otherwise, two triangles will not be formed upon division). It can pass through vertex $B$ or another vertex. In the first case (where $BD$ is the cut line), at least one of the resulting triangles, for example, triangle $BDC$, will not be acute-angled, so $BC$ is its base. Then $\angle DBC = \angle DCB = \alpha$, and $\angle BDA = 2\alpha$ is the exterior angle of triangle $BDC$. In triangle $ABD$, side $AB$ cannot be the base (otherwise, from the equality $DA = DB = DC$, it would follow that $\angle B = 90^{\circ}$, which contradicts the condition). Therefore, its base is either $AD$ or $BD$.
![](https://cdn.mathpix.com/cropped/2024_05_06_d922855093f726234d59g-3.jpg?height=194&width=946&top_left_y=1025&top_left_x=634)

If $AD$ is the base, then $AB = DB$ (left figure). Then $\angle A = \angle BDA = 2\alpha$. From triangle $ABC$, we get $2\alpha + \alpha = 60^{\circ}$, so $\alpha = 20^{\circ}$. Thus, $\angle C = 20^{\circ}$ and $\angle A = 40^{\circ}$. If $BD$ is the base, then $AB = AD$ (right figure). Then $\angle ABD = \angle BDA = 2\alpha$. Therefore, $\angle B = 3\alpha$, so $\alpha = 40^{\circ}$, which means $\angle C = 40^{\circ}$. In the second case, the cut line must pass through the vertex of the larger of the two acute angles (where $AE$ is the cut line, see figure). Then both resulting triangles will be obtuse-angled. In this case, $AB = BE$ and $AE = EC$. Therefore, $\angle BAE = \angle BEA = \frac{1}{2}(180^{\circ} - \angle B) = 30^{\circ}$, and $\angle EAC = \angle ECA = \frac{1}{2} \angle BEA = 15^{\circ}$.

Thus, $\angle C = 15^{\circ}$ and $\angle A = 45^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d922855093f726234d59g-3.jpg?height=166&width=466&top_left_y=1479&top_left_x=1549)