Question

If the bisector of the pair alternate interior angle framed by a transversal with two given lines are parallel. Prove that given lines are parallel

Answer 1

Given: $AB$ and $CD$ are the bisector of $\angle PAC$ and $\angle RCA,AB\parallel CD$

Since $AB$ and $CD$ are angles bisectors

So, $\angle DCR=\angle DCA=\alpha$ and $\angle PAB=\angle CAB=\beta$

Now, $AB\parallel CD$

$\angle BAC=\angle DCA$

$\beta =\alpha$

Since, alternate angles are equal

Thus, $PQ\parallel RS$