Today I constructed a 30o angle using a compass and ‘straight edge’. The proof of why this works is shown above.
Start with an arbitrary line:
Draw a circle with centre ‘P’ that is somewhere near half way across the line, mine seems to look exactly half way, but that’s just coincidence! Then, keeping the compass width the same (for all circles hereafter), draw a second circle with centre Q:
Then a third circle (same radius) from point S (where the circles intersect):
Point Q is where our third circle intersects with our second circle. A straight line between P and S will form a 30o angle RPT:
Then I coloured it in using the four colour theorem just for fun.