Construction of Special Angles
Constructing a 60° angle
Step 1     Draw a line l and mark a point O on it.

Step 2     Place the pointer of the compasses at O and draw an arc of convenient radius which cuts the line  PQ at a point say, A.

Step 3    With the pointer at A (as centre), now draw an arc that passes through O.

Step 4     Let the two arcs intersect at B. Join OB. We get BOA whose measure is 60°.

Constructing a 30° angle
Construct an angle of 60° as shown earlier. Now, bisect this angle. Each angle is 30°, verify by using a protractor.

Constructing a 120° angle
An angle of 120° is nothing but twice of an angle of 60°. Therefore, it can be constructed as follows :
Step 1     Draw any line PQ and take a point O on it.

Step 2    Place the pointer of the compasses at O and draw an arc of convenient radius which cuts the line at A.

Step 3    Without disturbing the radius on the compasses, draw an arc with A as centre which cuts the first arc at B.

Step 4     Again without disturbing the radius on the compasses and with B as centre, draw an arc which cuts the first arc at C.

Step 5     Join OC, COA is the required angle whose measure is 120°.

Constructing a 90° angle
Step 1    With A as centre and any suitable radius draw an arc cutting AB at P.
Step 2    With P as centre and the same radius as before cut the arc of Step 1 at Q. With Q as centre and the same radius cut the arc again at R.
Step 3    With Q and R as centres and any convenient radius (same for both) draw arcs cutting at S. Join A to S and produce A to L. Then BAL = 90°, i.e., AL is perpendicular to AB at A.