I require to construct a reasonably accurate $9^{\circ}$. And quickly. I learn that the quickest way is to make a straight line of $180^{\circ}$, bisect it (to make $90^{\circ}$), bisect that angle (to create a $45^{\circ}$ angle), and 5-sect* it.
Using a straight edge and a compass, how would I do this accurately? I need the error margin to be below 0.01%.
Additionally, I need proof.
It is on paper, and my pencil is as sharp as possible.
*For extra brownie points, tell me the technical word for 5-secting an angle.
Edit: @kennytm got the brownie points !!
You do not quintisect any angles (which is impossible with unmarked straightedge and compasses and an open problem if you have a marked straightedge). Instead, use a more ingenious linear combination.
Study Ptolemy's construction of the regular pentagon given here. (This link is also the source of the illustration given below.)
Once you have constructed the pentagon (or just the needed parts to do the steps below), you then have various ways to render the $9°$ angle. Two are given below:
Angle $PCQ$ measures $36°$ and is bisected by diameter $CD$. Either bisect one of the resulting half-angles or subtract the full angle $PCQ$ from $45°$.
Draw chord $BC$. This makes a $45°$ angle with diameter $CD$, and the side from $C$ to the unlabeled vertex on the right makes a $54°$ angle with that diameter. The $9°$ angle is the difference between these two angles.