A regular 18-gon is dissected into 18 pentagons, each of which is congruent to pentagon as shown. All sides of the pentagon have the same length.
Determine angles and What is their sum?
Bonus: Is collinear?
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
We notice at the center of the 18-gon, six pentagons are joined together by their angle that corresponds to ∠ A ⟹ ∠ A = 6 3 6 0 ∘ = 6 0 ∘ .
Since all the sides of the pentagon are equal, △ A B E is equilateral and quadrilateral B C D E is a rhombus.
∠ A B C is an interior angle of the 18-gon, so ∠ B = ∠ A B C = 1 6 0 ∘ .
Then ∠ E B C = ∠ A B C − ∠ A B E = 1 6 0 ∘ − 6 0 ∘ = 1 0 0 ∘ .
which implies ∠ D = ∠ C D E = ∠ E B C = 1 0 0 ∘ and ∠ C = ∠ B E D = 1 8 0 ∘ − ∠ E B C = 1 8 0 ∘ − 1 0 0 ∘ = 8 0 ∘ .
Finally, ∠ E = ∠ A E D = ∠ A E B + ∠ B E D = 6 0 ∘ + 8 0 ∘ = 1 4 0 ∘ .
To summarize: ∠ A = 6 0 ∘ , ∠ B = 1 6 0 ∘ , ∠ C = 8 0 ∘ , ∠ D = 1 0 0 ∘ , ∠ E = 1 4 0 ∘
⟹ S = 6 0 ∘ + 1 6 0 ∘ + 8 0 ∘ + 1 0 0 ∘ + 1 4 0 ∘ = 5 4 0 ∘ .
For the Bonus question, the answer is yes they are collinear but requires to prove ∠ X Y Z = 1 8 0 ∘ . (I will provide the proof later)