Question

What is the measure of a central angle of the regular polygon of which each interior angle is 140°?

Answer 1

40

Explanation:

Answer 2

`\underset{in~degrees}{\textit{Sum of All Interior Angles}}\\\\ n\theta = 180(n-2) ~~ \begin{cases} n=\stackrel{number~of}{sides}\\ \theta = \stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ \theta =140 \end{cases}\implies n140=180(n-2) \\\\\\ 140n=180n-360\implies 0=40n-360\implies 360=40n \\\\\\ \cfrac{360}{40}=n\implies 9=n`

so the polygon has 9 sides, that means a circumscribing circle will have 9 equal central angles, and since a circle has a total of 360°, that means each central angle will be 360 ÷ 9 = 40°.