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Given a diagram of n infinitely long straight lines in the plane, let them intersect in the points p_i, let the angles at $p_i$ be $v_i^j$, such that $360=\sum_j v_i^j$.

Given the suggestive diagram, and some angles, if one is given the task of calculating some $v_m^n$, and the diagram and angles determine this angle uniquely, is it always sufficient for finding this angle to solve the linear system $360=\sum_j v_i^j$ for every i, together with $a+b+c=180$ for every triangle?

One cant read any distances or other things off the diagram, only the order in which line a intersects line b, for every a, for every b.

What if one is given some additional information, that some distances are equal? Is it necessary to draw any more straight lines? Does one never need any more advanced formula/geomtry to find $v_n^m$?

(Why) does the algorithm fail for http://www.cut-the-knot.org/triangle/80-80-20/60-70Sol1.shtml#solution

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How would your question be answered in the case of two lines (one point of intersection)? –  hardmath May 10 '12 at 20:00
    
Yes, we need this rule too, angle on opposite side=equal –  user1708 May 10 '12 at 20:02
    
In mathematics (as opposed to art), lines are by definition infinitely long and straight. You could thus simplify the premise: "Given $n$ lines in the plane, ...". –  Théophile May 10 '12 at 21:34
    
You may have missed the point of the @hardmath comment. If all you know is that there are 2 lines, you can write down all the equations for angles you like, but you'll never be able to solve them, since the angle between the two lines could be anything (between zero and ninety degrees). –  Gerry Myerson May 11 '12 at 6:19
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Just think about the case of two lines. Of course it's not possible to deduce the angle in some way; the problem has too many parameters, and you don't have enough information; no algorithm is going to work; all the trigonometric theorems in all the libraries in the world aren't going to help. If all you know is you have two lines, how in the name of all that's mathematical are you going to calculate the angle between them? You need more information about the lines. Maybe you have more information about the lines, but if so you are keeping it to yourself. –  Gerry Myerson May 11 '12 at 13:34

1 Answer 1

Let's assume the lines are never parallel (so each pair of lines intersects once), and that no three lines are coincident at a point (so the number of intersections is exactly the number of pairs of lines).

Each line after the first introduces a new undetermined "angle" parameter. In the case of two lines, you have one undetermined value (since the four angles of intersection are all mutually determined by any one of them). As each new line is introduced, think about slightly wiggling the angle or slope of the line, so little that the order of the intersections along that line does not change. This is the new undetermined "angle" parameter that goes with the new line.

So fixing the angle between the first two lines would do little to help determine the angles introduced by adding later lines.

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