Mathematics
In the given circle with centre O, angle ABC = 100°, ∠ACD = 40° and CT is a tangent to the circle at C. Find ∠ADC and ∠DCT.
Answer
From figure,
⇒ ∠ADC + ∠ABC = 180° [Sum of opposite angles in a cyclic quadrilateral = 180°]
⇒ ∠ADC + 100° = 180°
⇒ ∠ADC = 180° - 100°
⇒ ∠ADC = 80°.
In △ADC,
⇒ ∠ADC + ∠CAD + ∠ACD = 180° [By angle sum property of triangle]
⇒ 80° + ∠CAD + 40° = 180°
⇒ ∠CAD = 180° - 120° = 60°.
From figure,
⇒ ∠DCT = ∠CAD = 60° [Angles in alternate segment are equal].
Hence, ∠DCT = 60° and ∠ADC = 80°.
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