To get the angle between the vector (1,1,1) and the plane z = 0, you can enter
angle between vectors (1,1,1) and (1,1,0)
or, using the WL built-in function,
VectorAngle[{1, 1, 1}, {1, 1, 0}]
The solutions are given in radians. I did not succeed to get the expression in degrees using only natural language, but you can mix with WL functions as
VectorAngle[{1, 1, 1}, {1, 1, 0}] in degrees
otherwise in pure Wolfram Language, it would be
VectorAngle[{1, 1, 1}, {1, 1, 0}]*180/Pi
To get the three angles from a single request:
N[VectorAngle[{1, 1, 1}, #]&/@{{0,1,1},{1,0,1},{1,1,0}}]