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The angles A, B, C and D of a quadrilateral are in the ratio 2 : 3 : 2 : 3. Show this quadrilateral is a parallelogram.

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Given, angles of a quadrilateral are in the ratio 2 : 3 : 2 : 3

i.e. A : B : C : D are in the ratio 2 : 3 : 2 : 3

To prove – Quadrilateral ABCD is a parallelogram

Proof – Let us take ∠ A = 2x, ∠B = 3x, ∠C = 2x and ∠D = 3x

We know, that the sum of interior angle of a quadrilateral = 360°

⇒ ∠A + ∠B + ∠C + ∠D = 360°

⇒ 2x + 3x + 2x + 3x = 360°

⇒ 10x = 360°

⇒ x = 360°/10 = 36°

∴ ∠A = ∠C = 2x = 2 × 360° = 72°

∠B = ∠D = 3x = 3 × 36° = 108°

Now, A quadrilateral ABCD is considered as a parallelogram.

(i) When opposite angles are equal,

i.e.  ∠A = ∠C = 72° and ∠B = ∠D = 108°

(i) When opposite angles are equal,

i.e. ∠A = ∠C = 72° and ∠B = ∠D = 108°

(ii) When adjacent angles are supplementary

i.e. ∠A + ∠B = 180°

and ∠C = ∠D = 180°

⇒ 72° + 108° and 72° + 108° = 180°

⇒ 180° = 180° and 180° = 180°

Since, quadrilateral ABCD fulfils the conditions

∴ Quadrilateral ABCD is a parallelogram.

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