The link in the answer by Aquatris is not working for me.
The angle between any two lines is given by the dot product.
Define each line segment as a vector from one point to the next in the point sequence, i.e. segment D1 = P2 - P1 = {P2x-P1x, P2y-P1y, P2z-P1z}. Likewise, define segment D2 as P3 - P2 = {P3x-P2x, P3y-P2y, P3z-P2z}. Now normalize these two vectors to make then unit vectors U1 = D1/|D1| and U2 = D2/|D2|.
Take the dot product of U1 and U2; this is the cosine of the angle between line segment 1 and 2.
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