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|  | [ The sum of the angles of a triangle. Theorem about the exterior angle. ] |  |
|  | [ Criteria and properties of an isosceles triangle. ] |  |
|  | [ Isosceles, inscribed, and circumscribed trapezoids ] |  |

In an isosceles triangle $ABC (AB = BC)$, the bisector $BD$ is twice as short as the bisector $AE$. Find the angles of triangle $ABC$.
Let $\angle A=2 \alpha$. Extend triangle $A B C$ to rhombus $A B C F$. $B F=2 B D=A E$, therefore, trapezoid $A B E F$ is isosceles. Thus,
$\angle A F E=\angle B A F=4 \alpha$, and $\angle B F E=\angle B A E=\alpha$. Therefore, $\angle A B F=\angle A F B=3 \alpha$. From the right triangle $A B D$ we get $2 \alpha+3 \alpha=90^{\circ}$, from which $\alpha=18^{\circ}$.

## Answer

$36^{\circ}, 36^{\circ}$ and $108^{\circ}$.