Why are right angles important I am interested if there is a concise mathematical way of expressing what is important about a right angle.
I am not so much asking for, say, a list of applications of right angles. Obviously they are used in endless situations and analyses. But my gut feeling is that there is something fundamental about the concept which might be able to be expressed concisely or even beautifully.
It may be that the answer I'm looking for is just the concept of orthogonality. But I'm not a mathematician and so I'm unsure if that is really a fundamental concept itself or actually a kind of result derived from something more basic.
 A: Right angles give us a convenient system of orthogonality, that helps us break down bigger things into components that can be analyzed independently. Think of how in physics when we calculate "work-done", we can neglect all components of a force which are orthogonal to the direction of displacement. 
When we started the primitive business of measuring things, we encountered tons of objects which stood "perpendicularly" on the ground, in some loose sense of the word. Tall trees, hills, take your pick. People realized that taller objects create larger shadows, and at some point, this correlation led them to the question of this relation could be used to measure how tall things are. 
If a $1$m stick creates a $10$cm shadow at afternoon, how tall is the huge tree which has a $10$m shadow at the same point of the day? This gave rise to trigonometry. People built homes, pyramids, so on and so forth using all these clever techniques.
A: From a definition standpoint, Euclid found a beautiful way to define right angles:

When a straight line intersects another straight line such that the adjacent angles are equal to one another, then the equal angles are called right angles and the lines are called perpendicular straight lines.

(definition taken from this website).
This definition is really a wonderfully efficient way to pin down the concept of a right angle! In order to think about angles, we need some concept of "the space between two lines," however we choose to interpret that. But once we have two lines (or rays, line segments, whatever) that intersect, the bare minimum requirement to talk about angles, we can define right angles using only the concept of equality, with no reference to numbers even. 
A: I will not attempt to give a concise way of expressing what is important about a right angle. Rather I can give a famous example:
Right-angled triangles are important because of Pythagoras Theorem $a^2+b^2=c^2$. This includes famous problems about right-angled triangles in number theory, for example the congruent number problem, relating to elliptic curves, Fermat's last Theorem and the Birch-Swinnerton-Dyer conjecture.
