10,11

Find the dihedral angles of the pyramid $ABCD$, all edges of which are equal to each other.

#
Let all edges of the given pyramid be equal to $a$. Since the lateral edges $D A, D B$, and $D C$ of the pyramid $A B C D$ are equal, its height $D O$ passes through the center $O$ of the circumscribed circle of the equilateral triangle $A B C$, i.e., through the center of this triangle. Let $M$ be the midpoint of $A B$. Then $C M$ is the height and median of the equilateral triangle, and since $D M$ is the height and median of the isosceles triangle $A D B$, $D M C$ is the linear angle of the dihedral angle at the edge $A B$ of the given pyramid. Denote $\angle D M C=\alpha$. From the right triangle $D O M$ we find that

![](https://cdn.mathpix.com/cropped/2024_05_06_acc13965d0370bfc03dcg-22.jpg?height=112&width=620&top_left_y=2628&top_left_x=719)

It is clear that the other dihedral angles of the given pyramid are also equal to $\alpha$.

## Answer

Problem