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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 - Mathematics

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.

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Solution

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°

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