What's the chance of an acute triangle?

[superbonzo]

... because both the set of all triangles and the set of acute triangles have the same cardinality, since there exits a corresponding bijection, it doesn't make sense to compare this sets in terms of percentage/ratio? (Same goes for the set of obtuse triangles, or any subset of the set of all triangles on which a similar bijection may be defined.)

Not exactly. As I sead in post #19, the concept of "numbers that count things" is modeled as cardinal numbers. Cardinal numbers are "usual" integers only for finite sets, thus the usual notion of "ratio" can be defined only for them.

Now, you can do two things:

1) try defining what is a ratio of general cardinal numbers; you can do that excatly in the same way you construct the set of rational numbers from the set of natural numbers ( with some added tecnicalities, it all depends on the axiomatic system you choose... ).
But as you noted, cardinality is a very coarse grained way of looking at sets: the interval [0,1] is the same as the interval [0,0.5], but it's also the same as any subset of R^n, like the interior of a square, a cube or a 100-dimensional sphere...

2) conclude that probability has to do with ratios and countings only when finite sets are involved. With infinite sets you need different concepts. The most common way of representing a probabilistic model is through a "probability space" ( that in turn is a special case of a measure space ) and of "random variables" ( that are simply measurable functions between probability spaces ).

In this way you can see the interval [0,1] as a probability space and you'll be able to conclude that the measure (ie the probability) of the interval [0,0.5] is 0.5.

But, is there some property of these sets that would enable as to quantify the relation between a set of all triangles and it's subset of acute triangles?

yes, you can define a probability structure over the set of triangles ( as an exotic example, you can send bijctively each triangle to a point in the interval [0,1] and use its natural probability measure directly ).

What you cannot do is defining a translation invariant probability measure over that set, because such a thing does not exist ( eg. there's no such probability space in the class of probability spaces ).