The area of a trapezium is 180 cm

Let the lengths of the parallel sides be $x \mathrm{~cm}$ and $(x+6) \mathrm{cm}$.

Now,

Area of trapezium $=\left\{\frac{1}{2} \times(x+x+6) \times 9\right\} \mathrm{cm}^{2}$

$=\left(\frac{1}{2} \times(2 x+6) \times 9\right) \mathrm{cm}^{2}$

$=4.5(2 x+6) \mathrm{cm}^{2}$

$=(9 x+27) \mathrm{cm}^{2}$

Area of trapezium $=180 \mathrm{~cm}^{2}$ (Given)

$\therefore 9 x+27=180$

$\Rightarrow 9 x=(180-27)$

$\Rightarrow 9 x=153$

$\Rightarrow x=\frac{153}{9}$

$\Rightarrow x=17$

Hence, the lengths of the parallel sides are $17 \mathrm{~cm}$ and $23 \mathrm{~cm}$, that is, $(17+6) \mathrm{cm}$.