Find the angle between the diagonal of the cube AC1 and the plane of the face ABCD.

Let the length of the edge of the cube be X cm.

We will construct the diagonal AC at the base of the cube and by the Pythagorean theorem, in the right-angled triangle ABC we will determine its length.

AC ^ 2 = AB ^ 2 + BC ^ 2 = X ^ 2 + X ^ 2 = 2 * X ^ 2.

AC = X * √2 cm.

Let’s construct a diagonal AC1.

The angle between the base plane and the AC1 diagonal is the linear angle CAC1.

In a right-angled triangle ACC1, tgCAC1 = CC1 / AC = X / X * √2 = √2 / 2.

Angle CAC1 = arctan (√2 / 2) ≈ 35.

Answer: The angle between the diagonal and the base plane is 35.