Is a 30 degree angle the opposite of a 60 degree angle?

My balls

Simple answer no.

You haven't specified whether the cut is

30° from the width (the 4 in 2×4)

or the cut is 30° with respect to length (example 8 feet).

flipping the board switches left and right but

not angle size.

A 90° turn of the board switches 30° for 60°, but

it's not clear how that would be useful.

30/60/90 triangles have interesting properties because

they're half of equilaterals (3 equal sides).

Example: to cut a 30° angle on the length, use

a tape to measure a diagonal that's twice the width.

To cut a 30 on the width measure down to .58 of width.

Update if you need more.

The 30° angle is the complement of the 60°, not opposite. Your 2×4 question is not so clear to me. Try this instead:

Begin with a rectangular table top, and cut off one corner. If the cut meets one edge at 30°, then other end of the cut meets an adjacent edge at 60°. The triangle that was cut off has angles 30°, 60°, and 90°.

No, generally moving an object does not alter its geometry.

Unless you move it too hard of course. In that case the angle can be whatever you want.

no, it's still 30º but in the opposite direction.