**Q) **ABCD is a parallelogram. P is a point on side BC and DP when produced meets AB produced at L. Prove that:

(i)

(ii)

(iii) If LP : PD = 2 : 3, then find BP : BC

**Ans: **

**VIDEO SOLUTION**

**STEP BY STEP SOLUTION**

(i) Since DC ǁ AB (and AL)

Line CB cuts these parallel lines,

Therefore, ∠ DCB = ∠ CBL

∠ DCP = ∠ PBL

Line DL cuts the parallel lines CD and AL,

Therefore, ∠ CDL = ∠ ALD

∠ CDP = ∠ BLP

Δ DCP ~ Δ PBL

(ii) Line DL cuts the parallel lines CD and AL,

Therefore, ∠ CDL = ∠ ALD

∠ CDP = ∠ ALD (interior angles)

Since ABCD is a parallelogram, therefore

Therefore, ∠ DAB = ∠ DCP (Opposite angles)

Δ DAL ~ Δ DCP

(iii) Since Δ DCP ~ Δ PBL

Given that =

or PC = BP

Since BC = BP + BP

BC = BP