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.