Question

Prove 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 point contact at the center.

Answer 1

The radius is perpendicular to the tangent at the point of contact.

$\angle A=\angle B={90}^o$

$\angle A+\angle B+\angle P+\angle O={360}^o$

${180}^o+\angle P+\angle O={360}^o$

$\angle P+\angle O={180}^o$.