图示为一个正五边形和一个正八边形。计算 $x$ 的值。

1. Find the interior angle of the regular pentagon and the regular octagon.

For a regular pentagon,
$\text{interior angle}=\frac{5-2}{5}\times 180^\circ=108^\circ.$

For a regular octagon,
$\text{interior angle}=\frac{8-2}{8}\times 180^\circ=135^\circ.$

2. Find angle $a$ using the angle sum of the isosceles trapezium in the overlap.

The two angles at the octagon side of the isosceles trapezium are each $135^\circ$.

Since the angle sum of a quadrilateral is $360^\circ$ and the remaining two angles are equal,
$a=\frac{1}{2}\left(360^\circ-2\times135^\circ\right)=45^\circ.$

3. Use the pentagon angle to calculate $x$.

At the relevant vertex of the pentagon, the interior angle is split into $x^\circ$ and $a$.

$x=108^\circ-45^\circ=63^\circ.$