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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?

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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!!

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I remember that once (Maybe $2000$ years ago), I asked this question on Physics SE, and the title of question was:

Can Newton's laws of motion be proved (mathematically or analytically) or are they just axioms?

There are many things I learnt from answers there abut the most thing was:

Asking whether one can prove something is a meaningless question unless one specifies the axioms one is allowed to use in the proof.

Whenever we see something in mathematical context we have two choices:

$1$. Assume it to be an axiom/postulate.

$2$. Prove it.

The first thing is easy to do because you have to do nothing, but it has also got some disadvantages. I mean you cannot be assured after choosing something as axiom because it can be wrong too, that is why we select only those things as axioms/postulates which are obvious.

Now, when we try to prove something, we do it by taking help of previously proved things or established axioms/postulates.

Now, let's come to your question, you are asking for a proof of equality alternate angles and whatever..

Now, it is quite right to think that way because yes you can prove that things provided that what axioms/postulates you want to assume true before constructing a proof.

For example: You can prove that alternate angles are equal by assuming that angles between two parallel lines are supplementary and a straight lines is just $180$ degrees (In terms of angles ).

I shall let you conclude.

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Actually everything can't have a proof. The things those proof you are wanted can be proved but at least one of them must be taken as an axiom. After that observe the Image. If any one is known or a axiom.

$(1)$ Suppose Co-interior property is known then the sum of angles on the same side is $180^\circ$ if one angle is $\theta$ then the co-interior angle is $180^\circ-\theta$ then the corresponding angles must equal by using linear pair property. And when corresponding angles are equal then by using vertically opposite angle property the alternate angles must be equal.

$$**OR**$$

$(2)$ Suppose alternate angle property is known the by using vertically opposite angle property corresponding angles are equal and then by using linear pair property the co-interior angle property be proved.

$$**OR**$$

$(3)$ Suppose corresponding angle property is known then by using the vertically opposite angle property the alternate angle must equalize and then by using the linear pair property the co-interior angle property can be proved.

Observe the Image

SORRY FOR ANY GRAMMATICAL MISTAKE.

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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.

Suggest you take advantage of the wealth of knowledge on WWW

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  • $\begingroup$ In the high school geometry text by Edwin Moise and Floyd Downs, they prove that if two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. They use this result to prove analogous results for corresponding angles and same-side interior angles. They then prove that if two parallel lines are cut by a transversal, then alternate interior angles are congruent. They use this result to prove that if parallel lines are cut by a transversal, then corresponding angles are congruent and same-side interior angles are supplementary. $\endgroup$
    – N. F. Taussig
    Mar 1, 2015 at 19:54
  • $\begingroup$ Are they starting the sequence of proofs without the corresponding angles postulate? $\endgroup$
    – Moti
    Mar 1, 2015 at 22:10
  • $\begingroup$ They start with the theorem that if one pair of alternate angles are congruent, then the other pair must be congruent. They prove that if a pair of alternate interior angles are congruent, then the lines are parallel by proving that if the lines are not parallel, then the other pair of alternate interior angles is not congruent. For the converse, they assume that alternate interior angles are not congruent, then derive a contradiction by showing that this implies that there are two lines parallel to the given line through a point not on the line. $\endgroup$
    – N. F. Taussig
    Mar 1, 2015 at 22:16
  • $\begingroup$ Any theorem requires a proof. I am wondering if they use only the Parallel Axiom or also use a "postulate". It seems that they replaced the corresponding with alternate congruency assumption or proved the alternate congruency based on the corresponding. I hope you see that alternate congruency is not prooveable without "help". $\endgroup$
    – Moti
    Mar 1, 2015 at 22:28
  • $\begingroup$ They only use the Parallel Axiom. $\endgroup$
    – N. F. Taussig
    Mar 1, 2015 at 22:30

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