The angles of a quadrilateral taken in order are 3$x$, 6$x$, 5$x$ and 4$x$.
Find the value of $x$. Answer $x$ = ___ Explain whether the quadrilateral is a parallelogram or a trapezium. because ___

1. Form an equation using the sum of angles in a quadrilateral and solve for $x$.

The angles in a quadrilateral add up to $360^\circ$.

$3x+6x+5x+4x=360$

$18x=360$

$x=20$

1. Find the actual angles of the quadrilateral using $x=20^\circ$.

From part (a), $x=20^\circ$.

The angles are
$3x=60^\circ,\quad 6x=120^\circ,\quad 5x=100^\circ,\quad 4x=80^\circ.$

2. Decide whether the quadrilateral has one or two pairs of parallel sides.

The angles are in order: $60^\circ,120^\circ,100^\circ,80^\circ$.

One pair of adjacent interior angles is supplementary:
$60^\circ+120^\circ=180^\circ.$
This shows one pair of opposite sides is parallel.

The other relevant pair is not supplementary:
$120^\circ+100^\circ=220^\circ \ne 180^\circ,$
so the other pair of opposite sides is not parallel.

Therefore, the quadrilateral is a trapezium, not a parallelogram.