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  • If in Figure bisectors AP and BQ of the alternate interior angles are parallel, then show that l || m.

If in Figure bisectors AP and BQ of the alternate interior angles are parallel, then show that l \parallel m.

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Solution.

In the figure AP \parallel BQ, AP and BQ are the bisectors of the alternate interior angles \angle CAB and \angle ABF.

To show : l \parallel m
Proof: Since AP \parallel BQ  and t is transversal therefore
\angle PAB = \angle ABQ (Alternate Interior Angles)
\Rightarrow 2 \angle PAB = 2\angle ABQ (Multiplying both sides by 2)
Now, AP and BQ are the bisectors of alternate interior angle \angle CAB and \angle ABF
2 \angle PAB = \angle CAB
2\angle ABQ = \angle ABF
So,
\angle CAB = \angle ABF
Now consider lines l and m
\angle CAB = \angle ABF (alternate interior angles are equal)
Hence l \parallel m

Hence proved

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