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Question
In a cyclic quadrilateral ABCD, ∠A : ∠C = 3 : 1 and ∠B : ∠D = 1 : 5; find each angle of the quadrilateral.
Solution
Let ∠A and ∠C be 3x and x respectively
In cyclic quadrilateral ABCD,
∠A + ∠C = 180°
(Pairs of opposite angles in a cyclic quadrilateral are supplementary)
`=>` 3x + x = 180°
`=> x = (180^circ)/4 = 45^circ`
∴ ∠A = 135° and ∠C = 45°
Let the measure of ∠B and ∠D be y and 5y respectively
In cyclic quadrilateral ABCD,
∠B + ∠D = 180°
(Pair of opposite angles in a cyclic quadrilateral are supplementary are supplementary)
`=>` y + 5y = 180°
`=> y = (180^circ)/6 = 30^circ`
∴ ∠B = 30° and ∠D = 150°
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