Question

PQR is a triangle formed by the adjacent sides PQ and QR and diagonal PR of a parallelogram PQRS. If in ΔPQR, ∠P : ∠Q : ∠R = 3 : 8 : 4, Calculate the measures of all the angles of parallelogram PQRS.

Answer 1

PQRS is a parallelogram.
Let ∠RPQ = 3x°
Then, ∠PQR = 8x° and ∠QRP = 4x°
In ΔPQR,
∠RPQ + ∠PQR + ∠QRP = 180°  ...(sum of angles of triangle= 180°)
3x° + 8x° + 4x° = 180°
15x° = 180°
x = 12°
⇒ ∠RPQ = 3x° = 3 x 12° = 36°
⇒ ∠PQR = 8x° = 8 x 12° = 96°
⇒ ∠QRP = 4x° = 4 x 12° = 48°
Now,
∠PSR = ∠PQR = 96° ...(opposite angles of a parallelogram are equal)
∠RPS = ∠QRP = 48° ...(Alternate angles since QR || PS)
∠PRS = ∠RPQ = 36° ...(Alternate angles since QR || PS)
Therefore,
∠PSR = ∠PQR = 96°, ∠RPS + ∠RPQ = 84°, ∠QRP = 84°.