there ar two points D and E on side AB of triangle ABC such that AD=BE. IF DP IS PARALLEL TO BC AND EQ IS PARALLEL TO AC, THEN PROVE THAT PQ IS PARALLEL TO AB

given: ABC is a triangle. AD = BE.

DP || BC and EQ || AC

TPT: PQ || AB

proof:

in the triangles ADP and EBQ;

AD = BE (given)

∠DAP = ∠BEQ [corresponding interior angles]

∠ADP = ∠EBQ [corresponding interior angles]

therefore by ASA congruency triangle ΔADP ≡ ΔEBQ

thus by CPCT : PD = BQ..........(1)

and PD || BQ [given]......(2)

since one pair of opposite side are equal and parallel.

therefore quadrilateral DPQB is a parallelogram and

 PQ || DB

i.e. PQ || AB

which is the required result.

hope this helps you.

cheers!!