Real World math question: given two poles of 3 m length, how do I easiest determine if they are parallel and aligned?
By "aligned" I mean that you can draw a rectangular square by connecting the starting points of the two poles and connecting the end points of the two poles. It is just a side effect of having same length and being parallel.
I currently do a cross measure between the diagonals and create two triangles. Using Pythagoras I can determine if they are aligned and thus in parallel.
Any other ideas?
If you just want to see if the poles are parallel, measure or compare the other two sides of the quadrilateral made by the poles. Those other sides are equal if and only if the quadrilateral is a parallelogram, which is equivalent to the poles being parallel (if you already know they have equal lengths and do not themselves intersect: you say you know the first, and the second is trivial to check).
You can find if the two poles, which have equal length, are opposite sides of a rectangle (which I think is what you want) by also measuring the two diagonals, or comparing them with each other. The two diagonals of a parallelogram are equal if and only if the quadrilateral is a rectangle. See Theorem 6-5-2 in Geometry by Edward B. Burger et al.
In summary, you can check "aligned" by seeing that the two poles do not intersect, and measuring or comparing the lengths of the diagonals and of the other two sides of the quadrilateral. No Pythagoras or other computation is needed. You can "compare" two lengths by marking one length on a measuring rod and comparing it directly with the other length.