What is Congruent Angles in Parallel Lines ?

Postulate 4.1 (The Parallel Postulate)

Through a given point not on a line there is exactly one line parallel to the line.

Theorem

If two parallel lines are cut by a transversal, each pair of alternate interior angles are congruent.

Given: and intersects at P
Prove: 1 @ 2
Proof: (indirect proof)

Assume m1  m2. Then we can draw another line through P such that the 3 formed by and is congruent to 2. From Theorem 4.1, // . That means is parallel to two lines in the plane at point P. This violates the Parallel Postulate. So we conclude that our assumption is false. Therefore, 1 @ 2