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

Given

`{:(∠ABC = 100^circ),(∠ACD = 40^circ):}}`   ...(Given)

∠ABC + ∠ACD = 180°   ...(Opposite angles of a cyclic quadrilateral)

100° + ∠ADC = 180°

∴ ∠ADC = 180° – 100° = 80°

Also,

∠ACD + ∠ADC + ∠CAD = 180°  ...(Sum of angles of a triangle)

40° + 80° + ∠CAD = 180°

∠CAD = 180° – 120° = 60°

Now, ∠DCT = ∠CAD = 60°   ...(Alternate segment theorem)