4.15. Find the angle between the intersecting diagonals of adjacent faces of a cube.
4.15. Let the edge of the cube be $a$; we need to find the angle between the diagonals $A_{1} B$ and $B_{1} C$ (Fig. 4.13). This angle is equal to the angle between the lines $D_{1} C$ and $B_{1} C$, since $D_{1} C \| A_{1} B$. In $\Delta B_{1} D_{1} C$, we have $B_{1} D_{1}=D_{1} C=B_{1} C=a \sqrt{2}$, and therefore, $\angle B_{1} C D_{1}=60^{\circ}$.

Answer: $60^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-579.jpg?height=365&width=373&top_left_y=1477&top_left_x=46)

Fig. 4.14