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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 the trig solution | Comment 1 of 2
For simplicity let all the sides be of length 1 WLOG.

Distance AC can be found via the law of cosines:

d^2 = 1 + 1 - 2cos(136°) 
d ~= 1.85436770913357

sin(BCA) ~= sin(136°)/1.85436770913357 ~= .374606593415914
               and BCA is acute as ABC is obtuse in triangle ABC

BCA ~= 22.0000000000001°, assume = 22°

Angle ACD is therefore (104 - 22)° = 82°

The distance AD is

(mAD)^2 ~= 1.85436770913357^2 + 1 - 2*1.85436770913357*cos(82°)
       ~= 3.92252339187661
       
mAD ~=  1.98053613748313

As AE and ED are each 1:

1.98053613748313^2 = 1 + 1 - 2cos(AED)

AED = arccos((1 + 1 - 1.98053613748313^2) / 2) = 164°

(calculator came up with 163.999999999996)

  Posted by Charlie on 2018-04-12 10:55:33
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