Question

# In the given figure, AB || CD and a transversal t cuts them at E and F respectively. If EP and FQ are the bisectors of ∠AEF and ∠EFD respectively, prove that EP || FQ .

Solution

## It is given that, AB || CD and t is a transversal. ∴ ∠AEF = ∠EFD           .....(1)         (Pair of alternate interior angles) EP is the bisectors of ∠AEF.        (Given) ∴ ∠AEP = ∠FEP = $\frac{1}{2}$∠AEF ⇒ ∠AEF = 2∠FEP          .....(2) Also, FQ is the bisectors of ∠EFD. ∴ ∠EFQ = ∠QFD = $\frac{1}{2}$∠EFD ⇒ ∠EFD = 2∠EFQ         .....(3) From (1), (2) and (3), we have 2∠FEP = 2∠EFQ ⇒ ∠FEP = ∠EFQ Thus, the lines EP and FQ are intersected by a transversal EF such that the pair of alternate interior angles formed are equal.  ∴ EP || FQ        (If a transversal intersects two lines such that a pair of alternate interior angles are equal, then the two lines are parallel)MathematicsRS Aggarwal (2020, 2021)All

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