In the given figure, PQ, PR and ST are tangents to the same circle. If ∠P = 40° and ∠QRT = 75°, find a, b and c.

From figure,

PQ = PR [Tangents from a fixed point outside the circle are equal.]

⇒ ∠PRQ = ∠PQR = a [Since angle opposite to equal sides are equal]

In △PQR,

⇒ ∠QPR + ∠PQR + ∠PRQ = 180° [By angle sum property of triangle]

⇒ 40° + a + a = 180°

⇒ 2a = 180° - 40°

⇒ 2a = 140°

⇒ a = $\dfrac{140°}{2}$

⇒ a = 70°.

From figure,

⇒ ∠PRQ + ∠QRT + ∠TRS = 180° [Linear pair]

⇒ 70° + 75° + ∠TRS = 180°

⇒ ∠TRS = 180° - 145° = 35°

⇒ SR = ST [Tangents from a fixed point outside the circle are equal.]

⇒ ∠STR = ∠TRS = 35° [Since angle opposite to equal sides are equal]

∴ c = 35°

In △SRT,

⇒ ∠STR + ∠TRS + ∠TSR = 180° [By angle sum property of triangle]

⇒ 35° + 35° + b = 180°

⇒ b = 180° - 70° = 110°.

Hence, a = 70°, b = 110° and c = 35°.