Question

Find the angle between two radii at the centre of the circle as shown in the figure. Lines PA and PB are tangents to the circle at other ends of the radii and ∠APR = 140°.

Answer 1

\[\angle\]APR + 

\[\angle\] APB = 180º         (Linear pair)

\[\Rightarrow 140^o + \angle APB = 180^o\]
\[ \Rightarrow \angle APB = 40^o\]

Also,

\[\angle\]OAP =

\[\angle\]OBP = 90º      (Radius is perpendicular to the tangent at the point of contact)

In quadrilateral APBO, 

\[\angle APB + \angle OBP + \angle BOA + \angle OAP = 360^o\]
\[ \Rightarrow 40^o + 90^o + \angle BOA + 90^o = 360^o\]
\[ \Rightarrow \angle BOA = 140^o\]