If two parallel lines are cut by a transversal, then pair of interior angles are supplementary

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 interior angles on the same side of the transversal are supplementary. 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. Since, ∠ 6 and ∠ 5 form a linear pair, so we get: ∠ 6 + ∠ 5 = 180° ..... Statement (1) (see the following diagram) Also, we know that ∠ 6 and ∠ 3 are alternative interior angles, so we get: ∠ 6 = ∠ 3 ..... Statement (2) (see the following diagram) Now from statement (1) and (2), we get: ∠ 5 + ∠ 3 = 180° (see the below diagram) And similarly we can find: ∠ 6 + ∠ 4 = 180° (see the above diagram) Since, ∠ 5 + ∠ 3 = 180° and ∠ 6 + ∠ 4 = 180 and they are interior angles on the same side of the transversal, so the Fact is proved that, "If two parallel lines are cut by a transversal, then each pair of interior angles on the same side of the transversal are supplementary". 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 interior angles on the same side of the transversal is supplementary, then two lines are parallel to each other"