一个正多边形的内角为 140°。 求该多边形的边数。 图中显示了一个正八边形和等边三角形的草图。 计算 x。

1. Set up the formula for the interior angle of a regular polygon.

For a regular polygon with $n$ sides,
$\text{interior angle}=\frac{(n-2)\times 180^\circ}{n}$
Given the interior angle is $140^\circ$,
$\frac{(n-2)\times 180}{n}=140$

2. Solve the equation to find the number of sides.

$180n-360=140n$
$40n=360$
$n=9$

1. Find the interior angle of the regular octagon.

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

2. Use angles at a point to calculate $x$.

Each angle of an equilateral triangle is $60^\circ$.

Angles at a point add up to $360^\circ$, so
$x=360^\circ-135^\circ-60^\circ=165^\circ.$