18 Pentagons!

Geometry Level 2

A regular 18-gon is dissected into 18 pentagons, each of which is congruent to pentagon A B C D E ABCDE as shown. All sides of the pentagon have the same length.

Determine angles A , B , C , D , and E A, B, C, D, \text {and } E and What is their sum?

Bonus: Is X , Y and Z X, Y \text {and } Z collinear?

540 340 360 500 720

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1 solution

Hana Wehbi
Jun 9, 2018

We notice at the center of the 18-gon, six pentagons are joined together by their angle that corresponds to A A = 36 0 6 = 6 0 \angle A \implies \angle A= \frac{360^\circ}{6}=60^\circ .

Since all the sides of the pentagon are equal, A B E \triangle ABE is equilateral and quadrilateral B C D E BCDE is a rhombus.

A B C \angle ABC is an interior angle of the 18-gon, so B = A B C = 16 0 \angle B= \angle ABC= 160^\circ .

Then E B C = A B C A B E = 16 0 6 0 = 10 0 . \angle EBC=\angle ABC-\angle ABE= 160^\circ-60^\circ = 100^\circ.

which implies D = C D E = E B C = 10 0 \angle D= \angle CDE= \angle EBC= 100^\circ and C = B E D = 18 0 E B C = 18 0 10 0 = 8 0 \angle C= \angle BED=180^\circ -\angle EBC= 180^\circ - 100^\circ =80^\circ .

Finally, E = A E D = A E B + B E D = 6 0 + 8 0 = 14 0 . \angle E= \angle AED = \angle AEB+\angle BED= 60^\circ+ 80^\circ=140^\circ.

To summarize: A = 6 0 , B = 16 0 , C = 8 0 , D = 10 0 , E = 14 0 \color{#D61F06}\angle A= 60^\circ,\color{#20A900} \angle B= 160^\circ,\color{#3D99F6} \angle C=80^\circ,\color{#69047E}\angle D= 100^\circ,\color{#EC7300}\angle E=140^\circ

S = 6 0 + 16 0 + 8 0 + 10 0 + 14 0 = 54 0 . \implies S= 60^\circ+160^\circ+80^\circ+100^\circ+140^\circ=\boxed{540^\circ}.

For the Bonus question, the answer is yes they are collinear but requires to prove X Y Z = 18 0 \angle XYZ= 180^\circ . (I will provide the proof later)

Actually, you can solve this faster using the fact that all convex pentagons have the sum of their angles as 540 degrees, but kudos to you for rather elegantly solving for all of the individual angles of the pentagon.

Nick Turtle - 3 years ago

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The question has two parts, the first part is what is the measure of each angle, definitely their sum is 540 540 , maybe l should change the problem a bit.

Hana Wehbi - 3 years ago

The sum of the internal angles of any n-gon is 180(n-2). Thus for any pentagon the answer is 540.

Steven Adler - 4 months, 2 weeks ago

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That's true but I was asking for the measure of each angle first, then their sum should be 540 as an extra verification.

Hana Wehbi - 4 months, 2 weeks ago

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