KöMaL 2023 C1784

The shorter leg of right triangle $ABC$ has length $1$. The ratio of the angles $\varphi$ and $\varepsilon$ between the height from the right angle (perpendicular to hypotenuse $AB$) and the angle bisectors of the acute angles is $\frac{\varphi}{\varepsilon}=\frac{4}{5}.$ Find the angles of the triangle and the length of the height corresponding to the hypotenuse.