Question

In the given figure, two circles intersect each other at points S and R. Their common tangent PQ touches the circle at points P, Q.
Prove that, ∠ PRQ + ∠ PSQ = 180°

Answer 1

It is given that two circles intersect each other at points S and R.
Join RS.

The angle between a tangent of a circle and a chord drawn from the point of contact is congruent to the angle inscribed in the arc opposite to the arc intercepted by that angle.
PQ is the tangent to the smaller circle and PR is the chord.
∴ ∠RPQ = ∠PSR      .....(1)
Also, PQ is the tangent to the bigger circle and RQ is the chord.
∴ ∠RQP = ∠QSR      .....(2)
Adding (1) and (2), we get
∠RPQ + ∠RQP = ∠PSR + ∠QSR
⇒ ∠RPQ + ∠RQP = ∠PSQ             .....(3)       
In ∆PRQ,
∠RPQ + ∠RQP + ∠PRQ = 180º      .....(4)      (Angle sum property)
From (3) and (4), we get
∠PSQ + ∠PRQ = 180º
Hence proved.