**l, m and n are three parallel lines intersected by transversals p and q such that l, m and n cut … n cut off equal intercepts DE and…**

###### Answers:

Though E, draw a line parallel to p intersecting L at G and n at H respectively.

Since l | | m ⇒ AG | | BE

and AB | | GE [by construction]

∴ Opposite sides of quadrilateral AGEB are parallel .

∴ AGEB is a parallelogram .

Similarly , we can prove that BEHC is a parallelogram .

Now, AB = GE [opposite sides of | | gm AGEB]

and BC = EH [opposite sides of | | gm BEHC]

But, given that AB = BC . Thus, GE = EH

Now, △DEG and △FEH, we have

∠DEG = ∠FEH [vertically opposite angles]

GE = EH [proved above]

and ∠DGE = ∠FHE [alternate interior angles]

By ASA congruence axiom, we have

△DEG ≅ △FEH

Hence, DE = EF