In Δ ABC and Δ PQR,

∠ ABC ≅ ∠ PQR, seg BD and

seg QS are angle bisector.

`If (l(AD))/(l(PS)) = (l(DC))/(l(SR))`

Prove that : Δ ABC ∼ Δ PQR

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#### Solution

Proof : `(l(AD))/(l(PS)) = (l(DC))/(l(SR))` ∴ `(l(AD))/(l(DC)) = (l(PS))/(l(SR))`

According to angle bisector theorem, `(l(AD))/(l(DC)) = (l(AB))/(l(BC)) ; (l(PS))/(l(SR)) = (l(PQ))/((QR))`

`∴ (l(AB))/(l(BC)) = (l(PQ))/(l(QR))` and ∠ ABC ≅ ∠ PQR ..... (Given)

Δ ABC ∼ Δ PQR ........ (SAS Test)

Concept: Property of an Angle Bisector of a Triangle

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