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In the Given Circle with Center O, ∠Abc=100°, ∠Acd=40° and Ct is Tangent to the Circle at C. Find ∠Adc and ∠Dct. - Mathematics

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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.

Solution

∠ 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°

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