Question

In the given figure, AB is a common tangent of two circles intersecting at C and D. Write down the measure of ∠ACB + ∠ADB and justify it.

Answer 1

Use of Tangent–Chord Theorem

AB is a common tangent at A to the left circle and at B to the right circle.

At point A:

The angle between tangent AB and chord AC equals the angle in the opposite arc:
∠CAB = ∠CDB.

At point B:

The angle between tangent AB and chord BC equals the angle in the opposite arc:
∠CBA = ∠CDA.

Relation of the Two Angles in the Figure

In triangle ACB,

∠ACB = 180° − (∠CAB + ∠CBA).

Substitute using the tangent–chord theorem:

∠ACB = 180° − (∠CDB + ∠CDA)

= 180° − ∠ADB

(since ∠CDB + ∠CDA together form ∠ADB).

Add the Angles

Now add ∠ACB and ∠ADB:

∠ACB + ∠ADB

= [180° − ∠ADB] + ∠ADB

= 180°.