PQRS is a parallelogram whose diagonals intersect at M.if ∠PMS=54∘, ∠QSR=25∘ and ∠SQR=30∘ ;find : (i) ∠RPS (ii) ∠PRS (iii) ∠PSR.
Give : || gm PQRS in which diagonals PR & QS intersect at M.∴ ∠PMS=54∘ ; ∠QSR=25∘ and ∠SQR=30∘
To find : (i) ∠RPS (ii) ∠PRS (iii) ∠PSRProof : ∵ QR || PS⇒ ∠PSQ=∠SQR (Alternate ∠s)But ∠SQR=30∘ (Given)∴ ∠PSQ=30∘In ΔSMP,∠PMS+∠PSM+∠MPS=180∘or 54∘+34∘+∠RPS=180∘∠RPS=180∘−84∘=96∘Now ∠PRS+∠RSQ=∠PMS∴ ∠PRS+25∘=54∘ ∠PRS=54∘−25∘=29∘∠PSR = ∠PSQ+∠RSQ=30∘+25∘=55∘Hence (i) ∠RPS=96∘ (ii) ∠PRS=29∘ (iii) ∠PSR=55∘