# Theorem 4 â€“ Converse of Pair of Alternate Angles are Equal

**Theorem 4**

If a transversal intersects two lines in such a way that a pair of alternate interior angles are equal, the two lines are parallel.

**Given: ** A transversal EF intersects two lines AB and CD at M and N respectively such that âˆ 4 and âˆ 6 are a pair of alternate interior angles and âˆ 4 = âˆ 6.

**To Prove: ** AB | | CD

**Proof: ** Since CD and EF intersect each other at N.

âˆ´ âˆ 6 = âˆ 8 â€¦ (Vertically opposite angles)

But âˆ 4 = âˆ 6 â€¦ (Given)

âˆ´ âˆ 4 = âˆ 8

But these are a pair of corresponding angles (âˆ 1 and âˆ 5)

âˆ´ AB | | CD â€¦ (Corresponding angles axiom)

If two parallel lines are intersected by a transversal, show that the bisectors of any pair of alternate interior angles are parallel.

**Given: ** AB || CD and EF is transversal. PG is the bisector of âˆ AGH and HQ is the bisector of âˆ DHG.

**To Prove: ** GP || HQ

**Proof: ** AB || CD and EF is the transversal

âˆ´ âˆ AGH = âˆ DHG â€¦. (Alternate angles)

âˆ´ âˆ PGH = âˆ ASH (PG is the bisector of âˆ AGH)

âˆ QHG = âˆ DHG (HQ is the bisector of âˆ DHG.)

âˆ´ âˆ PGH = âˆ QHG â€¦(âˆ 1 = âˆ 2 and âˆ 3 = âˆ 4)

But these are alternate interior angles

GP || HQ.