Question

In given fig., quadrilateral PQRS, side PQ || side SR, AR = 5 AP, then prove that, SR = 5PQ

Answer 1

Proof:

side PQ || side SR and seg SQ is their transversal.   ...[Given]

∴ ∠QSR ≅ ∠SQP    ...[Alternative angles]

∴ ∠ASR ≅ ∠AQP   ...(i) [Q−A−S]

 In ΔASR and ΔAQP,

∠ASR ≅ ∠AQP   ...[From (i)]

∠SAR ≅ ∠QAP    ...[Vertically opposite angles]

∴ ΔASR ∼ ΔAQP   ...[AA test of similarity]

∴ `(AR)/(AP) = (SR)/(PQ)`   ...(ii) [Corresponding sides of similar triangles]

But, AR = 5 AP   ...[Given]

∴ `(AR)/(AP) = 5/1`   ...(iii)

∴ `(SR)/(PQ) = 5/1`   ...[From (ii) and (iii)]

∴ SR = 5PQ