 Hello and welcome to the session I am Deepika here. Let's discuss a question which says from that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at this center. Now we know that the lengths of tangents drawn from an external point to a circle are equal. So this is a key idea behind our question. We will take the help of this key idea to solve the above question. So let's start the solution. Now we are given that and Pb are two tangents drawn from an external point P to a circle is center of angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the center that is angle aob angle apb are supplementary that is aob plus angle apb is equal to 180 degree. Proof. Join op in triangles oap and obp we have a is equal to pb because according to our key idea the lengths of tangents drawn from an external point to a circle are equal again oa is equal to ob radii of the same circle op is equal to op common. Therefore triangle oap is congruent to triangle obp by sss congruency condition pa equal to angle opb and angle aop is equal to angle bop by cpcp is equal to angle opa plus angle opb. Now these two are equal. Therefore angle pb is equal to twice angle opa. Similarly equal to angle aop plus angle bop and these two are also equal by cpcp so angle aob is equal to twice angle aop. In triangle aop we have angle aop plus angle opa plus angle oap is equal to 180 degree because the sum of the three angles of a triangle is 180 degree. Now we know that the tangent at any point of a circle is perpendicular to the radius to the point of contact. Therefore angle oap is 90 degree. Therefore angle aop plus angle opa plus 90 degree is equal to 180 degree because the tangent at any point of a circle perpendicular to the radius through the point of contact. Therefore angle aop is equal to 180 degree minus 90 degree plus angle opa angle aop is equal to 90 degree minus angle opa this implies twice angle aop is equal to twice angle 90 degree minus angle opa. Now twice angle aop is equal to angle aob. Therefore angle aop is equal to 180 degree minus twice angle opa is angle apb this implies aob plus angle apb is equal to 180 degree. Hence aob and angle apb are supplementary. Hence proved I hope the solution is clear to you. Bye and take care.