Question

The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. Show that the quadrilateral is a trapezium.

Answer 1

Let the angles of an quadrilateral be 3x, 4x, 5x and 6x .
We know that sum of four angles of quadrolateral is ${360}^{\circ }$
$\therefore 3x+4x+5x+6x={360}^{\circ }18x={360}^{\circ }x={20}^{\circ }3x=3\times {20}^{\circ }={60}^{\circ }4x=4\times {20}^{\circ }={80}^{\circ }5x=5\times {20}^{\circ }={100}^{\circ }6x=63\times {20}^{\circ }={120}^{\circ }LetA={60}^{\circ },\angle B={80}^{\circ },\angle C={100}^{\circ }and\angle D={120}^{\circ }\angle A+\angle D={60}^{\circ }+{120}^{\circ }={180}^{\circ }\angle B+\angle C={80}^{\circ }+{100}^{\circ }={180}^{\circ }$
Now A and D are supplementary
and B and C are supplementary
$\therefore AB||CD$
When 2 opposite sides are parallel, the quadrilateral is a trapezium.