In parallelogram PQRS. O is the mid-point of SQ. Find ∠S, ∠R. PQ, Qr and diagonal PR.

Given:

∠RQY = 60°

∵ ∠RQY and ∠RQP form a linear pair

⇒ ∠RQP = 180° - ∠RQY = 120°

∴ ∠RQP =120°

Now, in a parallelogram, the opposite sides are equal,

⇒ ∠RQP = ∠S = 120°

Similarly, ∠P = ∠R

Now, from the angle sum property of a quadrilateral,

∠P + ∠R + ∠Q + ∠S = 360°

⇒ ∠P + ∠R + 120° + 120° = 360°

⇒ 2∠P = 120°

⇒ ∠P = 60°

⇒ ∠R = 60°

Now,

Opposite sides of the parallelogram are equal.

⇒ SR = PQ = 15 cm

Also, PS = QR = 11 cm

And, the diagonals of the parallelogram bisect each other.

Hence, PR = 2×PQ = 2×6 = 12 cm.