Question

in the given figure , bisectors AP and BQ of the alternate interior angles are parallel, then show that l parallel to m​

Answer 1

Given:

AP and BQ are bisectors

∠EAB and ∠ABH are a pair of alternate interior angles.

To prove : l parallel to m

Solution

$\angle \: EAP \: = \angle \: PAB \: and \: \angle \: ABQ \: = \angle \: QBH ......( \: i \: )$

AP parallel to BQ

HENCE, T is transversal line

$\implies \angle \: PAB \: = \angle \: BAQ \: \: (alt. \: ext. \: angle)......( \: i \: )$

Now,

From eq ( i )

$\angle \: EAB \: = \angle \: ABH$

$\implies \: \angle \: EAP \: + \angle \: PAB \: = \angle \: ABQ \: + \angle \: QBH$

$\implies \: 2 \angle \: PAB \: = 2\angle \: ABQ$

Both the 2 will get cancelled

$\angle \: PAB \: = \angle \: ABQ$

$\therefore \: \angle \: EAB = \angle \: ABH .......(iii)$

From the figure and eq (iii) we can say that

t is a transversal line intersecting l and m such that alt ang are equal

Thus l and m are parallel line

$\boxed {Note : if \: the \: alt \: angles \: are \: equal \: then \: the \: the \: lines \: are \: parallel}$

Answer 2

Answer:

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