Chord to Tangent

A more general result concerns the angles formed by any chord with an endpoint on the point of tangency:

If line RS is tangent to circle O at P and PA is a chord, then the measure of angle SPA is one-half the measure of arc PA and the measure of angle RPA is one-half the measure of arc PBA.

 

We can see why this is true as follows.

Draw diameter PC

 

Then angle CPA is one-half the measure of arc AC.  Since PC is perpendicular to the tangent line, angle SPA is complementary to this angle.  Furthermore, the measure of arc PA is 180o minus the measure of arc CA since these arcs form a semicircle.  Putting this together,

As for angle RPA, it is supplementary to angle SPA, and the measure of arc PA is 360o minus the measure of arc PBA, so