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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.


Solution

Join AB

PQ is the tangent and PA is a chord

QPA=PBA.....(i) (angles in alternate segment)

similarly,

PQA=QBA....(ii)

Adding (i) and (ii), QPA+PQA=PBA+QBA

But in PAQ,QPA+PQA=180PAQ....(iii)

And PBA+QBA=PBQ....(iv)

from (iii) and (iv),

PBQ=1800PAQ

PBQ+PAQ=1800

Hence PBQ and PAQ are supplementary

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