Question

If angles $A$, $B$, $C$ and $D$ of the quadrilateral $ABCD$, taken in order, are in the ratio $3:7:6:4$, then $ABCD$ is a _______

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

Answer 1

The correct option is C

Trapezium

Explanation of the correct answer

Let the constant of proportionality be $x$

Hence the angles of the quadrilateral are $3x,7x,6x$ and $4x$.

According to the angle sum property of a quadrilateral

$3x+7x+6x+4x={360}^{\circ }$

$\Rightarrow$ $20x={360}^{\circ }$

$\Rightarrow$ $x={18}^{\circ }$

Hence the angles of the quadrilateral are ${54}^{\circ },{126}^{\circ },{108}^{\circ },{72}^{\circ }$ respectively

The opposite angles are equal in a parallelogram and rhombus. Here no two angles are equal.

So the quadrilateral cannot be a rhombus or a parallelogram.

Now, $\angle A+\angle B={54}^{\circ }+{126}^{\circ }={180}^{\circ }$

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

Thus angles on the same side of the transversal add up to ${180}^{\circ }$.

Hence lines $AD\parallel BC$

Thus the quadrilateral has one pair of parallel lines.

Hence, the given quadrilateral is a trapezium.