Question

Prove that The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Answer 1

Data:

A straight line SAT touches a given circle with centre O at A. AC is a chord through the point of contact A. ∠ADC is an angle in the alternate segment to ∠CAT and ∠AEC is an angle in the alternate segment to ∠CAS.

To Prove:

(i) ∠CAT = ∠ADC

(ii) ∠CAS = ∠AEC

Construction:

Draw AOB as diameter and join BC and OC.

Proof: