Dividing in Polar form
It sounds like you don't know that polar form is actually shorthand for exponential form of complex numbers.

Complex numbers can be expressed in three equivalent forms: Polar, exponential, trigonometric cartesian form, and simple cartesian form (from left to right).

$$
M \angle \theta = Me^{j\theta} = M(cos \theta +jsin \theta) = x + jy
$$

Polar form is just a shorthand form of exponential so you can write things more cleanly (it doesn't have any real mathematical meaning on its own).

Trigonometric Cartesian form just the expanded form of simple Cartesian form.

If you want to know why exponential form is equal to cartesian form, you should look it up. In words, when you expand the exponential form using the Taylor expansion. Each term has a j to which either an odd or even exponent is applied. The even terms cause the j to disappear while the odd terms cause j to remain. What's left is a group of real terms and a group of imaginary terms. The real terms happen to be the Taylor expansion of cos and the imaginary terms happen to be the Taylor expansion of sin. It's magical. I recommend you just take a glance at the math. That's enough for you to know in the back of your mind where it comes from.

Anyways, so when you divide this term, you divide the magnitude M as normal. You also divide the exponential as normal. However, when you divide exponentials you subtract the power:

$$
\frac{A \angle \theta} {B \angle \delta} =  \frac{A e^{j \theta}} {B e^{j \delta}} = \frac{A}{B} e^{j( \theta - \delta)} = \frac{A}{B} \angle{(\theta - \delta)} 
$$