What makes us say that Pythagoras theorem can be used in higher dimensions too? Pythagoras theorem seems to be a geometric property of our Universe. It's a property that helps us find the distances between two points in coordinate geometry in one dimension, two dimensions and three dimensions. 
But what makes us comment that this geometrical property can too be used in higher dimensions too.
 A: $\sqrt {\sqrt{x^2 + y^2}^2 + z^2} = \sqrt {x^2 + y^2 + z^2}$.  That's why.
A: It is just a theoretical extension of the three spatial dimensions in vector analysis.
So for $x \in \mathbb{R^n}$ the magnitude or length of a vector is $$| x |=\sum_{i=1}^n  x_i=\sqrt{x_1^2+x_2^2+x_3^2+\cdots+x_n^2}$$
A: "Property of our universe" - no.
Here is the thing. There is math, and there is empirical science. When you talk about math, it's somewhat "simple", you take formulas and derive more of them. Pythagoras theorem lives in math and can be extended there.
As for the universe, let me give you a related term here: euclidean distance. The euclidean distance between two points in an euclidean space is $d = \sqrt{\sum^n_{i=1}(x_i - y_i)^2}$. Sounds familiar, right?
Now realize that there are non-euclidean spaces, in which this does not hold, and in which pythagoras theorem also doesn't.
When you go with "property of your universe", what I hear is "something that we measured by empirical methods". We didn't. We evaluated that the theorem works, but it does not come from empirism.
And the most important thing: it is not true for our universe. It works for you when you measure the angles of a house or the like, but especially when you look at the General Theory of Relativity, you find the problems with this. Space-time is bent by mass and thus not flat.
For more on this, see https://www.quora.com/What-is-the-proof-that-space-isnt-Euclidean
Thus, this is not at all a property of our universe. This is a mathematical property of a mathematically defined space.
Oh, and if you consider time as a dimension, you need to go into Minkowski spaces, which makes things... complicated.
