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.
PQ is the tangent and PA is a chord
∠QPA=∠PBA.....(i) (angles in alternate segment)
Adding (i) and (ii), ∠QPA+∠PQA=∠PBA+∠QBA
But in △PAQ,∠QPA+∠PQA=180−∠PAQ....(iii)
from (iii) and (iv),
Hence ∠PBQ and ∠PAQ are supplementary