Can we construct $72^{\circ}$ with a compass and ruler?

Question

I wanted to know if it is possible to construct a $72^{\circ}$ angle via a ruler and compass? I know some numbers cannot be constructed using a ruler and compass, due to Galois Theory, but I don't exactly know how to check if a number falls in that category or not. Any resources would be appreciated as well!

Answer 1

This doesn't need Galois theory. We have $\cos(72^\circ)=\frac{\sqrt5-1}4$, so we can construct $72^\circ$ using the following steps:

1. Draw a line of length $\sqrt5$. (It is well-known that you can construct $\sqrt a$ from $a$)
2. Subtract $1$.
3. Divide by $4$. (Repeatedly bisecting; this creates $\cos(72^\circ)$)
4. Draw a right triangle with hypotenuse $1$ and adjacent $\cos(72^\circ)$. (In detail: Draw a perpendicular line from the line segment of length $\cos(72^\circ)$ drawn above, and intersect with a circle of length $1$.)
5. The angle the two lines above create is $72^\circ$.