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Consider the following diagram. There are two parallel lines with two transversals which cross at arbitrary angles (ignore any symmetry in the drawing).

enter image description here

From the "alternate interior angles" theorem, we can conclude that 1 = 2, 3 = 4, 5 = 6, and 7 = 8 by considering each transversal separately. However, can we say something about pairs of angles formed by different transversals? For example, can we say that 6 = 1 and 2 = 5? Intuitively, it makes sense, but I'm having a hard time proving it.

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    $\begingroup$ If they are "arbitrary angles", then no. $\endgroup$ May 29 '18 at 6:16
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enter image description here

does it make sense now? just consider the angles more arbitrary than the diagram you made

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