Question

if a transversal intersects two line such that a pair of alternate interior angles is equal then prove that the lines are parallel

Answer 1

For an alternate use see Corresponding angles (congruence and similarity) One pair of corresponding angles. With parallel lines, they are congruent.

Corresponding angles are the four pairs of angles that:

have distinct vertex points,lie on the same side of the transversal andone angle is interior and the other is exterior.

Two lines are parallel if and only if the two angles of any pair of corresponding angles of any transversal are congruent (equal in measure).

Note: This follows directly from Euclid's parallel postulate. Further, if the angles of one pair are congruent, then the angles of each of the other pairs are also congruent. In our images with parallel lines, corresponding angle pairs are: α=α1, β=β1, γ=γ1 and δ=δ1.