Question

The measures of the angles of a quadrilateral taken in order are in the ratio 1:2:3:4. The quadrilateral is a___

• A. Trapezium
• B. Parallelogram
• C. Rhombus
• D. Square

Answer 1

The correct option is A

Trapezium

Let ABCD be the given quadrilateral.

Let the angles of the quadrilateral be $x^{\circ },2x^{\circ },3x^{\circ }$ and $4x^{\circ }$

(sum of angles of a quadrilateral)

$x+2x+3x+4x={360}^{\circ }$

$10x={360}^{\circ }$

$x={36}^{\circ }$

$\therefore$ the $\angle A={36}^{\circ },\angle B=2\left({36}^{\circ }\right)={72}^{\circ },\angle C=3\left(36\right)={108}^{\circ }$ and $\angle D=4\left(36\right)={144}^{\circ }$

now $\angle B+\angle C=72+108={180}^{\circ }$

$\therefore AB||CD$ (Cointerior angles are supplementary)

Also, $\angle A+\angle B={36}^{\circ }+{72}^{\circ }={108}^{\circ }\ne {180}^{\circ }$

$\therefore$ AD is not parallel to BC.

$\therefore$ ABCD is a trapezium.