Question

To draw a pair of tangents to a circle which are inclined to each other at an angle 65 it is required to draw tangents at the end points of those two radii of the circle then find the angle BTW them

Answer 1

Answer:

Suppose that O be the center of the circle and  P and Q be any 2 points on the circle.

Draw tangents through P and Q such that two tangents meet at A outside the circle and angle PAQ=65

We know all tangents are perpendicular to radii at the points of contact

So OPAQ is a cyclic quadrilateral . The angles PAQ and POQ are supplementary.

Therefore angle POQ=180–65=115

Answer 2

see figure, here TP and TQ are two tangents are drawn from point T two circle at point P and Q respectively.

a/c to question, angle between two tangents are 65° e.g., $\angle{PTQ}=65^{\circ}$

we know, the tangent of the circle is perpendicular to its radius.
so, PT is perpendicular to OT and TQ is perpendicular to OQ.
e.g., $\angle{TPO}=\angle{TQO}=90^{\circ}$

from, quadrilateral TPOQ,
$\angle{PTQ}+\angle{TPO}+\angle{TQO}+\angle{QOP}=360^{\circ}$

$\implies65^{\circ}+90^{\circ}+90^{\circ}+\angle{QOP}=360^{\circ}$

$\angle{QOP}=115^{\circ}$