#### QUESTION

#### Ratio of consecutive angles of a quadrilateral is 1:2:3:4. Find the measure of its each angle. Write, with reason, what type of a quadrilateral it is.

#### SOLUTION

Let ₹PQRS be the quadrilateral.

Ratio of consecutive angles of a quadrilateral is 1 : 2 : 3 : 4.

Let the common multiple be x.

∴m∠P = x°, m∠Q = 2x°, m∠R = 3x° and m∠S = 4x°

In ₹PQRS,

m∠P + m∠Q + m∠R + m∠S = 360°

…[Sum of the measures of the angles of a quadrilateral is 360°]

∴x° + 2x° + 3x° + 4x° = 360°

∴10 x° = 360°

∴x° = 36010

∴x° = 36°

∴m∠P = x° = 36°

m∠Q = 2x° = 2 × 36° = 72°

m∠R = 3x° = 3 × 36° = 108° and

m∠S = 4x° = 4 × 36° = 144°

∴The measures of the angles of the quadrilateral are 36°, 72°, 108°, 144°.

Here, m∠P + m∠S = 36° + 144° = 180°

Since, interior angles are supplementary,

∴side PQ || side SR

m∠P + m∠Q = 36° + 72° = 108° ≠ 180°

∴side PS is not parallel to side QR.

Since, one pair of opposite sides of the given quadrilateral is parallel.

∴The given quadrilateral is a trapezium.

Let ₹PQRS be the quadrilateral.

Ratio of consecutive angles of a quadrilateral is 1 : 2 : 3 : 4.

Let the common multiple be x.

∴m∠P = x°, m∠Q = 2x°, m∠R = 3x° and m∠S = 4x°

In ₹PQRS,

m∠P + m∠Q + m∠R + m∠S = 360°

…[Sum of the measures of the angles of a quadrilateral is 360°]

∴x° + 2x° + 3x° + 4x° = 360°

∴10 x° = 360°

∴x° =

∴x° = 36°

∴m∠P = x° = 36°

m∠Q = 2x° = 2 × 36° = 72°

m∠R = 3x° = 3 × 36° = 108° and

m∠S = 4x° = 4 × 36° = 144°

∴The measures of the angles of the quadrilateral are 36°, 72°, 108°, 144°.

Here, m∠P + m∠S = 36° + 144° = 180°

Since, interior angles are supplementary,

∴side PQ || side SR

m∠P + m∠Q = 36° + 72° = 108° ≠ 180°

∴side PS is not parallel to side QR.

Since, one pair of opposite sides of the given quadrilateral is parallel.

∴The given quadrilateral is a trapezium.