Consider the words typically associated with geometry. are there any words that would be hard to precisely define? what words can you think of?

These words related to geometry that are quite hard to define precisely are called the undefined terms. There are three undefined terms in geometry: the point, the line and the plane. They are called as such, not because you can't necessarily define them. Since these are the basic elements of geometry, they are used to define all other terms in geometry. 

However, you can still describe these three undefined terms. We describe point as an indication of location space. It can be dimensionless, represented using coordinates ans so much more. Lines can go on infinitely in two directions. They don't have any thickness. Planes are two-dimensional flat surfaces that extend indefinitely in all directions. So, you see, there are no specific definitions of these terms. It depends on where you use them. That's what makes it hard to define precisely.

M< 5 = 80 , < 2 = 80, < 6= 100, < 3= 100, < 7= 80, < 4= 80, < 8= 100

The answer would be 7 cages