In the figure, 𝑃𝑄 is a tangent to a circle with centre 𝑂. if ∠𝑂𝐴𝐵 = 30°, find ∠𝐴𝐵𝑃 and ∠𝐴𝑂𝐵.

We have been given ∠𝑂𝐴𝐵 = 30°and we need to find ∠𝐴𝐵𝑃 and ∠𝐴𝑂𝐵. Since, the tangent is perpendicular to the end point of the radius,

𝑂𝐴 = 𝑂𝐵

 ⇒ ∠𝑂𝐴𝐵 =∠𝑂𝐵𝐴 

∠𝑂𝐵𝐴 = 30° 

Now, since the sum of angles of a triangle is always equal to 180 ° 

∠𝐴𝑂𝐵 = 180 − (30° + 30° ) 

⇒∠𝐴𝑂𝐵 = 120° 

Also, ∠𝐴𝐵𝑃 = ∠𝑂𝐵𝑃 −∠𝑂𝐵𝐴 

⇒ ∠𝐴𝐵𝑃 = 90 −30 

⇒∠𝐴𝐵𝑃 = 60° 

Hence ∠𝐴𝐵𝑃 = 60° and ⇒∠𝐴𝑂𝐵 = 120°