10. In the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, let the left focus be $F$, the right vertex be $A$, and the upper endpoint of the minor axis be $B$. If the eccentricity of the ellipse is $\frac{\sqrt{5}-1}{2}$. Then $\angle A B F$ $\qquad$ .
10. $90^{\circ}$

Let $c$ be the semi-focal distance of the ellipse, then from $\frac{c}{a}=\frac{\sqrt{5}-1}{2}$ we get $c^{2}+a c-a^{2}=0$.
Also, $|A B|^{2}=a^{2}+b^{2},|B F|^{2}=a^{2}$,
so $|A B|^{2}+|B F|^{2}=2 a^{2}+b^{2}=3 a^{2}-c^{2}$.
And $|A F|^{2}=(a+c)^{2}=a^{2}+2 a c+c^{2}$
$$
=3 a^{2}-c^{2}=|A B|^{2}+|B F|^{2} \text {, }
$$

Therefore, $\angle A B F=90^{\circ}$.