Question

If a transversal intersects two parallel lines, then each pair of interior angles are supplementary. Prove the given statement.

Answer 1

Define interior angles and prove they are supplementary

Interior angles are the angles formed between two parallel lines on the same side of a transversal when a transversal crosses a pair of parallel lines.

Let $AB$ and $CD$ be two parallel lines and $EF$ be the transversal which intersects these lines at points $M,N$

In the figure $\angle 1,\angle 3$ and $\angle 2,\angle 4$ are the pairs of interior angles.

To Prove: $\angle 1+\angle 3={180}^{\circ }$ or $\angle 2+\angle 4={180}^{\circ }$

Proof:

As ray $ND$ stands on a straight line $EF$ the sum of the angles created by it is ${180}^{\circ }$

$\therefore$ $\angle 5+\angle 3={180}^{\circ }$

$\angle 5=\angle 1$ $(\because$corresponding angles$)$

$\therefore \angle 1+\angle 3={180}^{\circ }$

In a similar way it can also be proved that

$\angle 2+\angle 4={180}^{\circ }$

Hence , it is proved that each pair of interior angles is supplementary