Question

PQRS is a parallelogram. QA bisects LPQR and meets PS at A. A straight line through R and parallel to QA meets PQ produced at B and PS produced at C. Prove that Triangle PBC is an isoscles triangle.

PLease, Fast.....

Answer 1

Given,QA bisects ∠PQR,∴∠PQA=∠AQR...(i)Also,PA∥QR ∴∠AQR=∠PAQ...[alternate angles].....(ii)From (i) and (ii),∠PAQ=∠PQA.....(iii)In △PBC,AQ∥BC,∠PAQ= ∠PCB and∠PQA=∠PBC.....[Corresponding angles]Using (iii),we have∠PCB =∠PBCBut  ∠PBC and ∠PCB are two base angles of △PBC and they are equal.∴△PBC is an isosceles triangle.