The diagram shows a regular hexagon P and a regular n-sided polygon Q. Polygon Q has an exterior angle of 18°. Find the sum of angles a, b, c, d, e and f. Answer ... °

1. Find the interior angle of polygon Q from its exterior angle.

For a regular polygon, each interior angle and exterior angle are supplementary.

$\text{Interior angle of Q}=180^\circ-18^\circ=162^\circ$

2. Find each exterior angle of the regular hexagon at the four vertices not touching Q.

For a regular hexagon,

$\text{each exterior angle}=\frac{360^\circ}{6}=60^\circ$

So

$b=c=d=e=60^\circ$

3. Find the two top angles where the hexagon and polygon Q meet.

Each interior angle of a regular hexagon is

$180^\circ-60^\circ=120^\circ$

At each top vertex, angles around a point add to $360^\circ$.

$a=360^\circ-120^\circ-162^\circ=78^\circ$

Similarly,

$f=78^\circ$

4. Add the six required angles.

$a+b+c+d+e+f=78^\circ+60^\circ+60^\circ+60^\circ+60^\circ+78^\circ$

$=396^\circ$