How can I calculate alpha and beta in this specific case?

Start by writing Pythagoras' theorem in the triangle on the right. $$a^2=\left(\frac c2\right)^2+x^2$$ $$x^2=a^2-625$$ $$x=\sqrt{a^2-625}$$ Then use this $x$ for the triangle on the left: $$(h+x)^2+\left(\frac c2\right)^2=b^2$$ $$h^2+x^2+2hx+\left(\frac c2\right)^2=b^2$$ $$400+a^2-625+40\sqrt{a^2-625}+625=(80-a)^2$$ $$400+40\sqrt{a^2-625}=6400-160a$$ $$\sqrt{a^2-625}=15-4a$$ You can square it, find $a$, then $b$, and then $\alpha$ and $\beta$. If you want, you can get $x$ as well.