Changes

In quadrillateral APOQ ,<br>

In quadrillateral APOQ ,<br>

∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br>

∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br>

∠PAQ+∠POQ=<br>  

∠PAQ+∠POQ=<math>180^{0}</math><br>  

Or, ∠PAQ+∠POQ=<br>

Or, ∠PAQ+∠POQ=<math>180^{0}</math><br>

∠PAQ = -∠POQ ----------1<br>

∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br>

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

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

∠POQ+∠OPQ+∠OQP=<br>

∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br>

Or  ∠POQ+2∠OPQ=<br>

Or  ∠POQ+2∠OPQ=<math>180^{0}</math><br>

2∠OPQ=- ∠POQ  ------2<br>

2∠OPQ=<math>180^{0}</math>- ∠POQ  ------2<br>

From 1 and 2 <br>

From 1 and 2 <br>

∠PAQ=2∠OPQ

∠PAQ=2∠OPQ