in the given figure TAS is a tangent to a circle with centre O.at the point A, if angle OBA=32⁰. find the value of x?

in the given figure TAS is a tangent to a circle with centre O.at the point A, if angle OBA=32⁰. find the value of x?

As the figure is not very clear, if  TAS is the tangent to the circle at point A , and  A, B and C are the points on the circumference .

so the solution is:

∠OAB=90 deg. [line joining the point of contact to the centre is perpendicular to the tangent]

∠BAS=90-∠OAB=90-x

∠ACB=∠BAS [The angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment.]

∠ACB=90-x

∠AOB=2∠ACB=2(90-x)=180-2x  [the angle at the centre is twice the angle at the circumference]

now in triangle AOB,

∠OBA=32 deg, ∠AOB=180-2x , and ∠OAB=x

now since the sum of the interior angles in a triangle is equal to 180 deg.

OR

in the triangle OAB,

OA=OB= radius of the circle

therefore ∠OAB=∠OBA=32 deg.