Suppose that you have a line L and some point A on L:
How will you construct a ray (or line) through A which is inclined at 600 to L?
Step 1: Taking A as center and any radius, draw an arc which intersects L at B:
Step 2: Now, taking B as center and AB as radius, draw another arc which intersects the first arc at C:
Step 3: Draw a ray (or line) through A and C. This will be inclined at 600 to L:
Why does this construction work?
Proof: Note that AB = AC, since these are radii of the same circular arc. Also, BC = BA, since these too are radii of the same (second) circular arc. Thus,
AB = BC = AC
This means that \(\Delta ABC\) is equilateral, and so, \(\angle BAC\) = 600.
Note that you can construct an angle of 300 by bisecting an angle of 600, and you can further construct an angle of 150 by bisecting an angle of 300.