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What lines, segments, or rays are parallel in the following figures and what postulate or theorem proves them parallel? The figures are at this link http://media.cheggcdn.com/media/194/1941e517-b09f-4759-9cb1-6a033756def8/phpFZgyq2.png I think I have the first one solved. In the first figure I found segment DA to be parallel to segment CB and DC to be parallel to AB by the Consecutive Interior Angels Converse Theorem as they add up to 180 degrees so they are supplementary angles. I'm quite confused about the next two figures. I'd really appreciate some advice to get me going.

Mathematics
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Use the theorem that states "Two lines cut by a transversal with congruent interior angles are parallel." (12) And, for (13) use that the sum of two angles is the greater angle (as per Euclid's postulates) and the same case as (12).
Now, try *proving* the lines are parallel, or, better yet, to get an intuition, I'd recommend proving the actual theorems, if you do have the time. It'll really help out on getting the meat of why it works!
So for (12) segments HE and GF are parallel and because they are parallel the angles formed by the transveral are also parallel. Is that correct?

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Careful, HE is not parallel to GF, try finding out why. (Hint: check out the angles, and check triangles HFG and HEF. Try figuring out each of the angles.)
So the missing angles are 156 degrees and 88 degrees and the angles in both triangles amount to 180 as they are right triangles. I don't understand how that does not make them parallel.
Wait. Triangle HEF is the only right triangle. Triangle HFG does not have a right angle, right?
\[ m\angle EHF=180-90-34=56\ne 58 \]While: \[ m\angle FGH=180-58-34=88\ne 90 \]
Yes, that is correct. So, therefore, is it possible for HE to be parallel to GF?
No.
So if HE isn't parallel to GF then is HG parallel to EF? Are those two segments parallel to each other as the transveral cuts them?
Yes, now why? Try it out, consider the similar angles. And, I'm off, sorry, as I have to head to class.
Oh, I'm sorry for bothering you at such an important time. Thank you very much for helping me out so far. I appreciate it and I'll try my best on my own. Once again, thanks!
Which theorem best fits segments HG and EF in the following figure http://media.cheggcdn.com/media/194/1941e517-b09f-4759-9cb1-6a033756def8/phpFZgyq2.png ? The alternate interior angles theorem or the consecutive interior angles theorem?

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