Question

Two circles intersect each other at points A and B. Their common tangent touches the circles at points P and Q as shown in the figure. Show that the angle PAQ and PBQ are supplementary.

Answer 1

Join AB

PQ is the tangent and PA is a chord

$\angle QPA=\angle PBA$.....(i) (angles in alternate segment)

similarly,

$\angle PQA=\angle QBA$....(ii)

Adding (i) and (ii), $\angle QPA+\angle PQA=\angle PBA+\angle QBA$

But in $△PAQ,\angle QPA+\angle PQA=180-\angle PAQ$....(iii)

And $\angle PBA+\angle QBA=\angle PBQ$....(iv)

from (iii) and (iv),

$\angle PBQ={180}^0-\angle PAQ$

$\angle PBQ+\angle PAQ={180}^0$

Hence $\angle PBQand\angle PAQ$ are supplementary