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Angle in an Equilateral Pentagon (Posted on 2018-04-12) Difficulty: 4 of 5
Let ABCDE be an equilateral convex pentagon such that angle ABC = 136 and angle BCD = 104. What is the measure of angle AED?

An equilateral polygon is a polygon whose sides are all of the same length.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Comment 2 of 2 |
Without loss of generality we can assume that
the length of the sides of ABCDE is 1.
 
Applying x-y coordinates to the vertices of
ABCDE we have

   B [0,0]
   C [1,0]
   A [cos(136),sin(136)]
   D [1 + cos(76),sin(76)]

Point E must belong to the unit circles
with centers at A and D:

   Circle-A: [x - cos(136)]^2 +
             [y - sin(136)]^2 = 1

   Circle-D: [x - {1 + cos(76)}]^2 +
             [y - sin(76)]^2 = 1

There are two points that satisfy these
conditions. One makes ABCDE convex and
the other does not. The one that does -

   E [cos(76),sin(76)]

Clearly it satisfies Circle-D. I will let
you confirm that it satisfies Circle-A.

Clearly |BE| = 1. Therefore, triangle ABE
is equilateral and quadrilateral BCDE is
a rhombus. Therefore,

   angle(AEB) = 60 
   angle(BED) = angle(BCD) = 104

   angle(AED) = angle(AEB) + angle(BED) = 164.

QED

You can't construct this pentagon with
straightedge and compass.







  Posted by Bractals on 2018-04-12 12:38:48
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