Question

$AP$ and $BP$ are the tangents drawn from an external point $P$ to a circle with center $O$.

$\angle AOB$ and $\angle APB$ are always ________.

• A. complementary
• B. supplementary
• C. obtuse angles
• D. acute angles

Answer 1

Answer: (B)

supplementary

Tangent is perpendicular to the radius at the point of contact.

Therefore $\angle OAP=\angle OBP={90}^{\circ }$

We know that sum of all the four angles of a quadrilateral is ${360}^{\circ }$.

So in quadrilateral AOBP,

$\angle OAP+\angle AOB+\angle OBP+\angle APB={360}^{\circ }$

${90}^{\circ }+\angle AOB+{90}^{\circ }+\angle APB={360}^{\circ }$

$\angle AOB+\angle APB={180}^{\circ }$

Therefore $\angle AOB$ and $\angle APB$ are always supplementary.

So, option B is the answer.