Question

In the given circle with center o, ∠ABC=100°, ∠ACD=40° and CT is tangent to the circle at C. find ∠ADC and ∠DCT. 

 

Answer 1

In a cyclic  quadrilateral ABCD, 

`∠ ABC+∠ADC=180°`   (Opposite angle of a cydic quadrilateral are supplementary)

`⇒ 100°+∠ADC=180°`

`⇒ ∠ADC=80°`

Now, in `ΔACD`,

`∠ADC+∠CAD+∠ADC=180°`

`⇒40°+∠CAD+80°=180°`

`⇒∠CAD=180°-120°`

`⇒∠CAD=60°`

`"Now" ∠DCT=∠CAD ...........("angles in the alternate segment are equal")`

`∴ ∠DCT=60°`