Question

In the adjoining 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

Given: Two circles intersect each other at points S and R. 

line PQ is a common tangent. 

To prove: ∠PRQ + ∠PSQ = 180° 

Proof:

Line PQ is the tangent at point P and seg PR is a secant. 

∴ [∠RPQ = ∠PSR …………. (i) 

and ∠PQR = ∠QSR] ………… (ii) [Tangent secant theorem] 

In ∆ PQR, 

∠PQR + ∠PRQ + ∠RPQ = 180° [Sum of the measures of angles of a triangle is 180°]

∴ ∠QSR + ∠PRQ + ∠PSR = 180° [From (i) and (ii)] 

∴ ∠PRQ + ∠QSR + ∠PSR = 180° 

∴ ∠PRQ + ∠PSQ = 180° [Angle addition property