No, you can't quinsect any angle. Let's start from the fact that you can construct the pentagon (which is stated below). If you were able to quinsect its angle, you would have the 25-gon

Galois Theory (see any reference on finding the n-gon) tells us that we can construct the n-gon if and only if the number of invertibles in the ring Z/nZ is a power of 2. The invertibles in Z/5Z are Phi(5)=4, a power of 2.

The number of invertibles is given by the following formula:

Phi(p_1^n_1 * p_2^n_2* ... *p_k^n_k)=p_1^(n_1-1)*(p_1-1) * p_2^(n_2-1)* (p_2-1) * ... * p_k^(n_k-1)* (p_k-1)

where p_i are primes and n_i>0 are integers

So you can construct the n-gon with ruler and compass for the following values:

n=1,2,3,4,5,6,8,10,12,15,17, etc...

But for 25, you have Phi(5^2)=5*4, not a power of 2. You cannot get the 5-section of the 72degrees angle with ruler and compass.

So back to your question, How did you construct 9-gons (Phi(9)=3*2)?

There are methods that give approximate solutions for low n's, but they can never be exact in the case of the 9-gon or in the case of the pentagon.