## Task 3 - 290823

Let $A B C D E F G H$ be a cube with an arbitrary edge length (see figure).

a) Determine the size of the angle $\angle D E B$!

b) Prove that the angles $\angle A H B$ and $\angle B E C$ have the same size!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0793.jpg?height=299&width=355&top_left_y=316&top_left_x=1387)
a) The triangle $D B E$ is equilateral, as $D E, B E$, and $B D$ are face diagonals of the cube and thus equal in length. The angle we are looking for is an interior angle of this triangle, so $\angle D E B=60^{\circ}$.

b) The triangles $A H B$ and $B E C$ are congruent by the (sss) congruence theorem, since $B C=A B$ (edges of the cube), $B E=A H$ (face diagonals), and $C E=B H$ (space diagonals).

In these triangles, the angles we are looking for correspond to each other, as they are each enclosed by a face diagonal and a space diagonal. Therefore, $\angle A H B=\angle B E C$, q.e.d.