In the given figure, if OQ = QR, then the value of m is:

We already know that,

Opposite angles of equal sides of a triangle are equal

The sum of the interior opposite  angle of a triangle is equal to the exterior angle 

We have,

OQ = QR

∠QRO = n

∠POS = m

In Δ QOR

OQ = QR

So, ∠QOR = ∠QRO

= ∠QOR = ∠QRO = n      -----(a)

= ∠OQP = ∠QOR + ∠ORQ 

= ∠OQP = n + n      (from a)

= ∠OQP = 2n      -----(b)

In ΔOPQ

OQ = OP      (radii of the same circle)

= ∠OQP = ∠OPQ      -----(c)

From (2) and (3)

= ∠OPQ = 2n      -----(d)

In ΔPOR

= ∠POS = ∠PRO + ∠OPR  

= m = 2n + n

= m = 3n