# Constructing An Angle of 60 Degrees

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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.