In the given figure if ∠ATO=40°, find ∠AOB.

∠ATO = 40° …given

From T we have two tangents TA and TB

We know that if we join point T and centre of circle O then the line TO divides the angle between tangents

⇒ ∠ATO = ∠OTB = 40° …(i)

∠OAT = ∠OBT = 90° …radius is perpendicular to tangent …(ii)

Consider quadrilateral OATB

⇒ ∠OAT + ∠ATB + ∠TBO + ∠AOB = 360°…sum of angles of quadrilateral

From figure ∠ATB = ∠ATO + ∠OTB

⇒ ∠OAT + ∠ATO + ∠OTB + ∠TBO + ∠AOB = 360°

Using (i) and (ii)

⇒ 90° + 40° + 40° + 90° + ∠AOB = 360°

⇒ 260° + ∠AOB = 360°

⇒ ∠AOB = 100°