Changes

OP=OQ ----  radii of the same circle

OP=OQ ----  radii of the same circle

OA is joined.<br>

OA is joined.<br>

In quadrillateral APOQ ,

In quadrillateral APOQ ,<br>

∠APO=∠AQO= [radius drawn at the point of contact is perpendicular to the tangent]

∠APO=∠AQO= [radius drawn at the point of contact is perpendicular to the tangent]<br>

∠PAQ+∠POQ=  

∠PAQ+∠POQ=<br>

Or, ∠PAQ+∠POQ=  

Or, ∠PAQ+∠POQ=<br>

∠PAQ = -∠POQ ----------1

∠PAQ = -∠POQ ----------1<br>

Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP

Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br>

∠POQ+∠OPQ+∠OQP=

∠POQ+∠OPQ+∠OQP=<br>

Or  ∠POQ+2∠OPQ=

Or  ∠POQ+2∠OPQ=<br>

2∠OPQ=- ∠POQ  ------2

2∠OPQ=- ∠POQ  ------2<br>

From 1 and 2  

From 1 and 2 <br>

∠PAQ=2∠OPQ

∠PAQ=2∠OPQ