1. Draw any line PQ
2. With P as centre and any suitable radius, draw an arc to meet PQ at R.
3. With R as centre and same radius, draw an arc intersecting the previous arc at S.
4. With S as centre and same radius, draw another arc intersecting the arc at T.
5. With S and T as centre, draw arc of same radius to intersect each other at U.
6. Join PU, which is the required ray making 90o with given ray PQ.
7. Now, bisect angle UPQ, which gives two angles of measure 45o each.
8. Again, bisect angle 450 to get two angles measuring 22.50 each.
Now, 22.5o + 45o = 67.5o
Thus, angle APQ = 67.5o