Answer to Worlds Hardest Easy Geometry Problem

The problem is known as Langley's Adventitious Angles and was posed in 1922. It is also known as the hardest easy geometry problem because it can be solved by elementary methods but it is difficult and laborious.

Can you figure it out? Watch the video for a solution.

Can You Solve The Hardest Easy Geometry Problem?

Or keep reading for an explanation.
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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

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Answer To The Hardest Easy Geometry Problem

There are two main principles to solving the problem. The first is that all the angles in a triangle sum to 180 degrees. The second is that in an isosceles triangle, there are two equal angles opposite two equal sides. Knowing a triangle has two equal angles means the sides opposite are equal, and knowing there are two equal sides means the angles opposite are equal.

The proof involves working through a series of isosceles triangles. To get started, draw line segment BG such that CBG is equal to 20 degrees.

In triangle CBG, we know one angle is 20 degrees and other is 80 degrees, for a total of 100 degrees. So the triangle's angles sum to 180, we can solve that  ∠CGB = 80 degrees. This means triangle CBG is an isosceles triangle, and BC = BG.

Angles CBG and BGE form a straight line so they must add up to 180 degrees. This means angle BGE equals 100 degrees.

Then, focusing on triangle BGE, we can solve that ∠BEG = 40 degrees, because it has to be 180 minus the known angles of 40 and 100. Triangle BGE has two angles equal to 40 degrees, so this is another isosceles triangle, so BG = GE.

Then, focusing on triangle BFC, we can solve that ∠BFC = 50 degrees, which means triangle BFC is another isosceles triangle. This means BF = BC.

We have proven BC = BG = GE = BF.

Now we create another triangle BFG. Since BG = BF, we know the opposite angles must be equal. (If you are following along with the video, I misspoke this step in the video at 5:04. I meant to say sides BG and BF are equal.)

The third angle in the triangle, ∠GBF, is 60 degrees, so the remaining angles have to be half of 180 – 60. This is (180 – 60)/2 = 60 degrees. In other words, all 3 angles are equal so BFG is an equilateral triangle. All of its sides must be equal, so GF = BF.

We have figured out a lot of information. There is just one more triangle that is necessary to consider, so below is a diagram focusing on triangle GFE that omits the non-essential information.

We know GF = GE, so we once again have an isosceles triangle, and we know the vertex angle is equal to 40 degrees. This means the remaining angles are one-half of 180 – 40, which is 70 degrees.

Finally, we know that 40 + x has to be equal to 70, so that means x = 30 degrees.

And that's the answer! The value of x is 30 degrees.

Wikpedia Langley's Adventitious Angles
https://en.wikipedia.org/wiki/Langley%E2%80%99s_Adventitious_Angles

Math With Bad Drawings Solution
A Technique is Just a Trick That Went Viral

World's hardest easy geometry problem
http://thinkzone.wlonk.com/MathFun/Triangle.htm
https://www.duckware.com/tech/worldshardesteasygeometryproblem.html

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Source: https://mindyourdecisions.com/blog/2016/09/04/the-hardest-easy-geometry-problem-sunday-puzzle/

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