In the given figure, PT touches the circle with centre O at point R. Diameter SQ is produced to meet the tangent TR at P.
Given ∠SPR = x° and ∠QRP = y°;
(i) ∠ORS = y°
(ii) Write an expression connecting x and y.
`∠`QRP = `∠`OSR = y (angles in alternate segment)
But OS = OR (Radii of the same circle)
∴`∠`ORS = `∠`OSR = y
∴ OQ = OR (radii of same circle)
∴ OQR = `∠` ORQ= 90° - y …………(i) (Since OR ⊥ PT )
But in Δ PQR ,
Ext `∠` OQR = x + y ……(i)
From (i) and (ii)
x + y = 90°- y
⇒ x + 2 y = 90°