Question

In the given circle with centre O, ∠ABC = 100°, ∠ACD = 40° and CT is a tangent to the circle at C. Find ∠ADC and ∠DCT.

Answer 1

In cyclic quadrilateral ABCD,

∠B + ∠D = 180°   (opp angles of the cyclic quad are supplementary)

⇒ 100° + ∠ADC = 180°

=> ∠ADC = 80°

Now in ΔACD

∠ACD + ∠CAD + ∠ADC = 180°

40° + ∠CAD + 80° = 180°

∠CAD = 180° - 120° = 60°

Now ∠DCT = ∠CAD   (angles in the alternate segment)

∴ ∠DCT = 60°