Find the measure of each interior angle and each exterior angle of the following regular polygons. Show your work

[tex]\bf \textit{sum of all interior angles of a regular polygon}\\\\ n\theta =180(n-2)\qquad \begin{cases} n=\textit{number of sides}\\ \theta =\textit{interior angle} \end{cases}\\\\ -------------------------------\\\\ \textit{sum of all exterior angles of a regular polygon}\\\\ n\theta =360\qquad \begin{cases} n=\textit{number of sides}\\ \theta =\textit{interior angle} \end{cases}[/tex]

now, with that in mind... let's take a peek at .. hmmm say c) dodecagon
now, a Dodecagon has 12 sides.... let's check the angles then

[tex]\bf n\theta =180(n-2)\qquad n=12\implies 12\theta =180(12-2) \\\\\\ 12\theta =180\cdot 10\implies 12\theta =180\implies \theta =\cfrac{1800}{12} \\\\\\ \theta =150^o\impliedby \textit{interior angle of a Dodecagon}\\\\ -------------------------------\\\\ n\theta =360\qquad n=12\implies 12\theta =360\implies \theta =\cfrac{360}{12} \\\\\\ \theta =30^o\impliedby \textit{exterior angle of a Dodecagon}[/tex]

so... hmm now for a) a decagon is 10 sides, b) pentagon is 5 sides, d) and e) pretty much you can tell how many sides on those already.

eudora eudora · Answer 2 · 2019-06-10T19:19:38+02:00

Answer:

Step-by-step explanation:

As we know for a regular polygon sum of all interior angles is represented by n A = 180 (n-2)

and sum of all exterior angles is represented by n A' = 360°

Here n represents number of sides of the polygon.

Now we will go for each options given.

(A) Decagon : (Having 10 sides)

(1) 8A = 180 ( 8-2 )

8A = 180 × 6

A = [tex]\frac{1080}{8}[/tex] = 160° ( interior angle )

(2) n A' = 360°

8 A' = 360°

A = 45° ( exterior angle )

(B) Pentagon ( having 5 sides )

(1) 5A = 180 ( 5-2 )

5A = 180 × 3 = 540

A = 108° ( interior angle )

(2) 5A' = 360

A' = 72° (exterior angle )

(C) Dodecagon : ( having 12 sides )

(1) 12A = 180 (12-2)

12 A = 180 × 10 = 1800

A = 150° ( interior angle )

(2) 12 A' = 360

A' = 30° ( exterior angle )

(D) 16-gon ( having 16 sides )

(1) 16 A = 180 ( 16 - 2 )

16A = 180 × 14

16 A = 1520

A = 157.5° ( interior angle )

(2) 16 A' = 360

A' = 22.5°

(E) 25-gon ( having 25 sides )

(1) 25 A = 180 ( 25-2)

25A = 180 (23)

A = 165.6° ( interior angle )

(2) 25A' = 360

A = 14.4° ( exterior angle)