Question

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

Answer 1

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 angles of a quadrilateral = 360°

⇒ ∠A + ∠B + ∠C + ∠D = 360°

⇒ 2x + 3x + 2x + 3x = 360°

⇒ 10x = 360°

⇒ x = `360^circ/10 = 36^circ`

∴ ∠A = ∠C = 2x = 2 × 360° = 72°

∠B = ∠D = 3x = 3 × 36° = 360°

Now, A quadrilateral ABCD is considered as a parallelogram.

(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 fulfills the conditions.

∴ Quadrilateral ABCD is a parallelogram.

Answer 2

The sum of all angles in a quadrilateral is 360^∘. Let the angles of the quadrilateral be 2x, 3x, 2x, and 3x.

Using the sum of the angles:

2x + 3x + 2x + 3x = 360^∘.

Simplify: 10x = 360^∘

x = 36^∘

Substitute x = 36^∘ into the expressions for the angles:

• 2x = 2 × 36 = 72^∘
• 3x = 3 × 36 = 108^∘

Thus, the angles are: 72^∘, 108^∘, 72^∘, 108^∘

In a parallelogram, opposite angles are equal. Here:

• One pair of opposite angles: 72^∘, 72^∘,
• Another pair of opposite angles: 108^∘, 108^∘

Since opposite angles are equal, the quadrilateral satisfies the property of a parallelogram.

The given quadrilateral is a parallelogram because its opposite angles are equal