Question

how to prove that alternate angles are equal

Answer 1

the theorem
"If a transversal intersects two parallel lines, then each pair of alternate interior angles are equal. "
proves that alternate angles are equal

Answer 2

Take two parralel line segments.

Lets say AB & CD are parallel . & AB = CD

Now draw a transversal AD .

Now join AC & BD.

We find that traingle ACD is congruent to traingle BDC ( SSS ) .

BY CPCT we find alternate amgles are equal.

This is generalized as a theory . Because even if AB is not equal to CD , you can always construct this and you will get a rectangle with the transversal as its diagonal. In a rectangles the two triangles divided by diagonal are congruent . Thus alternate angles are equal.