If two parallel lines are cut by a transversal, then pair of alternate angles are equal

Transversal Property / Fact - (1)

Transversal Property / Fact - (2)

Transversal Property / Fact - (3)

If two parallel lines are cut by a transversal, then each pair of alternate angles has equal measure. Solution: In the below diagram Lines A and B are parallel to each other and a Transversal Z cuts lines A and B at two points thereby making different angles; marked as ∠ 1, ∠ 2, ∠ 3, ∠ 4, ∠ 5, ∠ 6, ∠ 7 and ∠ 8. Now, we know that vertically opposite angles are equal, so we get ∠ 7 = ∠ 6 ..... (statement 1) (Highlighted with yellow color in the below diagram) Since line A || B and Transversal Z cuts lines A and B at two points, so corresponding angles are equal and we get: ∠ 7 = ∠ 3 ..... (statement 2) (Highlighted with pink color in the below diagram) Now, from statement (1) and (2), we get ∠6 = ∠ 3 (Highlighted with blue color in the below diagram) Similarly, we can find; ∠ 5 = ∠ 4 (highlighted with yellow color in above diagram) Since, angles ∠ 6 & ∠ 3 and ∠ 5 & ∠ 4 are alternative interior angles, so the fact is proved that; "If two parallel lines are cut by a transversal, then each pair of alternate angles has equal measure". As we proved the above fact, so we can say that the converse of this above fact also hold true i.e. "If a transversal intersect two lines such that a pair of alternative angles is equal, then two lines are parallel to each other"