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In cyclic quadrilateral ABCD, ∠A = 3∠C and ∠D = 5∠B. Find the measure of each angle of the quadrilateral.

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Answer

Given, cyclic quadrilateral ABCD

In cyclic quadrilateral ABCD, ∠A = 3∠C and ∠D = 5∠B. Find the measure of each angle of the quadrilateral. Tangents and Intersecting Chords, Concise Mathematics Solutions ICSE Class 10.

So, ∠A + ∠C = 180° [Opposite angles in a cyclic quadrilateral is supplementary]

⇒ 3∠C + ∠C = 180° [As ∠A = 3∠C]

⇒ 4∠C = 180°

⇒ ∠C = 180°4\dfrac{180°}{4}

⇒ ∠C = 45°.

Now,

⇒ ∠A = 3∠C = 3 x 45° = 135°.

Similarly,

⇒ ∠B + ∠D = 180°

⇒ ∠B + 5∠B = 180° [As, ∠D = 5∠B]

⇒ 6∠B = 180°

⇒ ∠B = 180°6\dfrac{180°}{6}

⇒ ∠B = 30°.

Now,

⇒ ∠D = 5∠B = 5 x 30° = 150°.

Hence, ∠A = 135°, ∠B = 30°, ∠C = 45° and ∠D = 150°.

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