`AP` and `BQ` are the bisectors of the two alternate interior angles formed by the intersection of a transversal `t` with parallel lines `l` and `m` (in the given figure). Show that `AP|""|BQ`.

Explanation

Given In the figure l`|""|`m , AP and BQ are the bisectors of `/_`EAB and `/_`ABH, respectively.
To Prove AP`|""|` BQ
Proof Since, l`|""|`m and t is transversal.
Therefore,` " "/_EAB = /_ABH " " `[alternate interior angles]
`rArr " "1/2/_EAB = 1/2/_ABH " " `[dividing both sides by 2]
`rArr " "/_PAB = /_ABQ " " `[AP and BQ are the bisectors of `/_EAB and /_`ABH]
Since, `/_ `PAB and `/_`ABQ are alternate interior angles with two lines AP and BQ and transversal AB, Hence, AP `|""|` BQ.