Question

PQRS is a parallelogram whose diagonals intersect at M.
If ∠PMS = 54°, ∠QSR = 25° and ∠SQR = 30° ; find :

(i) ∠RPS
(ii) ∠PRS
(iii) ∠PSR.

Answer 1

Given : ||gm PQRS in which diagonals PR & QS intersect at M.
∠PMS = 54° ; ∠QSR = 25° and ∠SQR=30°

To find : (i) ∠RPS (ii) ∠PRS (iii) ∠PSR

Proof : QR || PS

⇒ ∠PSQ = ∠SQR (Alternate ∠s)

But ∠SQR = 30° (Given)

∠PSQ = 30°

In ∆SMP,

∠PMS + ∠ PSM +∠MPS = 180° or 54° + 30° + ∠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°