10,11

Through a point in space, four planes are drawn, no three of which have a common line. Into how many parts do these planes divide the space? What are the names of the resulting parts of space?
![](https://cdn.mathpix.com/cropped/2024_05_06_f7e07b07084714df1adcg-30.jpg?height=45&width=257&top_left_y=575&top_left_x=17)

A plane passing through the vertex of a trihedral angle and a point inside the trihedral angle divides the trihedral angle into a trihedral and a tetrahedral angle. Consider planes $\alpha$ and $\beta$, passing through a given point $O$. They divide the space into 4 parts. Draw a plane $\gamma$ through point $O$ and point $A$, not lying in planes $\alpha$ and $\beta$. Planes $\alpha, \beta$, and $\gamma$ divide the space into 8 trihedral angles with a common vertex $O$. Draw a plane $\delta$ through point $O$ and point $B$, not lying in any of the planes $\alpha, \beta$, and $\gamma$. We get four planes, no three of which have a common line. Plane $\delta$ passes through point $O$ and intersects exactly six of the eight trihedral angles formed by planes $\alpha, \beta$, and $\gamma$. Thus, six more parts are added to the already existing eight parts of the space. Therefore, there will be 14 parts in total, 8 of which are trihedral angles, and 6 are tetrahedral angles.

Btuppora itk: :ob.

Let the specified planes pass through point $O$. Consider four planes, each parallel to the given ones. They bound a certain tetrahedron. First, we prove that these planes divide the space into 15 parts. Let point $M$ be an arbitrary point in space, located outside the tetrahedron. Then the segment $MO$ intersects either a face of the tetrahedron (4 cases), or an edge (6 cases), or passes through a vertex of the tetrahedron (4 cases). Each case corresponds to exactly one (different from the interior of the tetrahedron) part of the space into which the space is divided by the specified planes. Thus, the total number of parts is $4+6+4+1=15$. This is what we needed to prove. The number of parts into which the four original planes divide the space is one less than the found number (the tetrahedron "turned into" a point, and one part disappeared). In this case, 6 parts of the space, which corresponded to the edges of the tetrahedron, turned into tetrahedral angles, and 8 parts, which corresponded to the vertices and faces, turned into trihedral angles.

## Answer

Into 14 parts, 8 of which are trihedral angles, and 6 are tetrahedral angles.