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Theorem 4 – 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 4 and 6 are a pair of alternate interior angles and 4 = 6.

To Prove: AB | | CD

Proof: Since CD and EF intersect each other at N.
6 = 8 … (Vertically opposite angles)
But 4 = 6 … (Given)
4 = 8
But these are a pair of corresponding angles ( 1 and 5)
AB | | CD … (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 AGH and HQ is the bisector of DHG.
To Prove: GP || HQ

Proof: AB || CD and EF is the transversal
AGH = DHG …. (Alternate angles)

PGH = ASH (PG is the bisector of AGH)
QHG = DHG (HQ is the bisector of DHG
PGH = QHG …( 1 = 2 and 3 = 4)

But these are alternate interior angles
GP || HQ.



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