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Question
ABCD is a cyclic quadrilateral in which BC is parallel to AD, angle ADC = 110° and angle BAC = 50°. Find angle DAC and angle DCA.
Solution
ABCD is a cyclic quadrilateral in which AD || BC
∠ADC = 110°, ∠BAC = 50°
∠B + ∠D = 180°
(Sum of opposite angles of a quadrilateral)
`=>` ∠B + 110° = 180°
`=>` ∠B = 70°
Now in ΔABC,
∠BAC + ∠ABC + ∠ACB = 180°
`=>` 50° + 70° + ∠ACB = 180°
`=>` ∠ACB = 180° – 120° = 60°
∵ AD || BC
∴ ∠DAC = ∠ACB = 60° ...(Alternate angles)
Now in ΔADC,
∠DAC + ∠ADC + ∠DCA = 180°
`=>` 60° + 110° + ∠DCA = 180°
`=>` ∠DCA = 180° – 170° = 10°
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