10. The four vertices of a regular tetrahedron \(ABCD\) lie on the same sphere, \(CC'\) and \(DD'\) are diameters of the sphere. Find the angle between the plane \(ABC\) and the plane \(AC'D'\).


10. With the edges of a regular tetrahedron as the face diagonals of a cube, the tetrahedron is inscribed in a cube, which is also inscribed in a sphere, and the diagonal of the cube is the diameter of the sphere. Connect \(C^{\prime} D^{\prime}\) intersecting \(A B\) at \(M\), connect \(M C\), then \(M C \perp A B, M D^{\prime} \perp A B\), so \(\angle C M D^{\prime}\) is the dihedral angle between the planes \(A B C\) and \(A C^{\prime} D^{\prime}\). Let the edge length of the cube be \(a\), in \(\triangle C M D^{\prime}\) we can find \(\sin \angle C M D^{\prime}=\frac{2 \sqrt{6}}{3}\). Therefore, the dihedral angle between the planes \(A B C\) and \(A C^{\prime} D^{\prime}\) is \(\arcsin \frac{2 \sqrt{6}}{3}\).