Question

The angles H, I, J and K of a quadrilateral are in the ratio 3:5:7:9. Then HIJK is a:

• A. kite
• B. parallelogram
• C. trapezium with HI II JK
• D. trapezium with IJ II HK

Answer 1

The correct option is C trapezium with HI II JK

Let the angles H, I, J and K of a quadrilateral in degrees be 3x, 5x, 7x and 9x respectively.

As we know, the sum of the measures of the angles of a quadrilateral is ${360}^{\circ }$.

Hence,
$\Rightarrow 3x+5x+7x+9x={360}^{\circ }$
$\Rightarrow 24x={360}^{\circ }$
$\Rightarrow x={15}^{\circ }$

Hence,
$\angle H={45}^{\circ },\angle I={75}^{\circ },\angle J={105}^{\circ }and\angle K={135}^{\circ }$

We can observe that,
$\angle H+\angle K={45}^{\circ }+{135}^{\circ }={180}^{\circ }$
$\angle I+\angle J={75}^{\circ }+{105}^{\circ }={180}^{\circ }$

$\angle H+\angle I\ne {180}^{\circ }$
$\angle J+\angle K\ne {180}^{\circ }$

Thus, $\angle H$ and $\angle K$ are adjacent and supplementary angles.
$\angle I$ and $\angle J$ are adjacent and supplementary angles.

Sum of adjacent angles on non-parallel sides of trapezium is ${180}^{\circ }$. (Property)

HI॥ JK (By the properties of supplementary angles)
Hence, HIJK is a trapezium with HI॥ JK.