Proof of Alternate, Corresponding and Co-interior Angles During school our teacher always explains the proof for all theorems even simple ones such as why does the angles in a triangle of add up to $180$ and they all involve alternate, corresponding or co-interior angles. However it has never occurred to me that he never shown the proof of how are alternate angles are equal. 

Could any explain the proof for alternate, corresponding and co-interior angles?
Not everything has a proof. As sawarnik link says that there are axioms and postulates regarding this fact. We havent been able to find 2 parallel lines with transversal who's alternate and co interior angles arent equal :)! 
The day someone does euclid and others will be proved wrong. But anyways u can always geometrically prove this by making a parallel line transversal setup. Cut the co interior angles briefly and putg them next to each other you will see that they will be exactly like a protractor or semi circle making 180° similarly corresponding  interior and alternative interior angles will overlap proving that they are equal...If they aren't you will become famous!!
The Corresponding Angles Postulate is the base for proving the other cases.

http://hotmath.com/hotmath_help/topics/corresponding-angles-postulate.html

Also go to: http://www.cliffsnotes.com/math/geometry/fundamental-ideas/postulates-and-theorems

As a postulate it does not have a proof and is part of the Parallel Lines axiom.

To summarize: based on the corresponding angles postulate you may prove the theorems with regard to alternate and co-interior angles. From there you derive the sum of the triangle angels to be 180.

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