Question

In the given figure, two circles intersect each other at points S and R. Their common tangent PQ touches the circle at points P, Q. Prove that, $\angle PRQ+\angle PSQ={180}^o$

Answer 1

REF.Image.

$\angle PRQ+\angle PSQ={180}^{\circ }$

$\angle SPQ=\frac{1}{2}m\left(arcPR\right)$

$\angle SQR=\frac{1}{2}m\left(arcPQ\right)$

Inscribed angle

$\angle PSQ+\angle PQS+\angle SQP={180}^{\circ }$ (sum of all angle)

$\angle SQR=\frac{1}{2}\times m\left(arcSRQ\right)$ also

$\angle SPQ=\frac{1}{2}m\left(arcSRP\right)$

But

$\angle PSQ=1/2\angle PRQ$

$arcPR+arcRQ={180}^{\circ }$

also

$\angle PRQ+\angle PSQ={180}^{\circ }$