We start with a line segment PQ that will be the
hypotenuse of the triangle..


First, we construct a 30° angle on one end of the hypotenuse. See
Constructing a 30° angle. 
1. Starting with the hypotenuse line PQ, set the compasses on P,
and set its width to any convenient width. 

2. Draw a broad arc across PQ. Label the point where it crosses PQ as point S. 

3. Without changing the compasses' width, move the compasses to the point S.
Draw a broad arc that crosses the first one.
Label the point where the two arcs cross as point T. 

4. Again without changing the compasses' width, move the compasses to the point T, and draw an arc across the previous arc,
creating point R. 

5. Draw a line from P through R and continue it on beyond R to create the second side of the triangle. 

Next, we construct a 60° angle on the other end of the hypotenuse. See
Constructing a 60° angle. 
6. Place the compasses on point Q and set it to any convenient width. 

7. Draw a broad arc across PQ on the same side as point R. Label the point where it crosses PQ as point A. 

8. With the compasses on A, draw a second arc, crossing the first arc at point B 

9. Draw a line from Q, through B and on to cross the line PR. Label the intersection point C. 

10. Done. The triangle PQC is a 306090 triangle. 
