# The World's Hardest Easy Geometry Problem



## smhbbag

World's Hardest Easy Geometry Problem

The directions are there, but I'll post them:



> Using only elementary geometry, determine angle x. Provide a step-by-step proof.
> 
> You may only use elementary geometry, such as the fact that the angles of a triangle add up to 180 degrees and the basic congruent triangle rules (side-angle-side, etc.). You may not use more advanced trigonomery, such as the law of sines, the law of cosines, etc. There is a review of elementary geometry below.
> 
> This is the hardest problem I have ever seen that is, in a sense, easy. It really can be done using only elementary geometry. This is not a trick question.



The figure is at the link.

Have fun. 

At first glance, my first thought was "that's trivial." But it's not. Many times I thought I had quick solutions, and I did not. Yall enjoy, and without wasting much more of my own time, I'm simply going to trust the mathematicians who have said elsewhere that it is genuinely hard. 

I feel like a sham of a math tutor, now.


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## smhbbag

bump for the evening crowd.

I looked up the solution online and it's.....ridiculous. I could've spent a few more days on it and never got it.


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## OPC'n

That's not what I call fun! You need to get out more!


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## BJClark

I am NOT a math person, so I passed it along to my daughter whom I know it will bug the day lights out of until she figures it out..she's trying to work it out now..


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## BJClark

She says it's X degrees..

I can delete the answer if others want to figure it out on their own...it took her less than 10 minutes..


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## Calvinist Cowboy

BJClark said:


> I am NOT a math person, so I passed it along to my daughter whom I know it will bug the day lights out of until she figures it out..she's trying to work it out now..



I guess I must be a math geek then. This was challenging!


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## BJClark

Calvinist Cowboy;




> I guess I must be a math geek then. This was challenging!



she didn't find it very challenging at all, she followed the first hint-- pulled out her protractor, had it solved in minutes..

Had I been trying to solve it, I'd be here for days...getting frustrated..not worth that to me..


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## smhbbag

> She says it's X degrees..
> 
> I can delete the answer if others want to figure it out on their own...it took her less than 10 minutes..



Protractors are cheating


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## Seb

I got the answer...but I had to use AutoCad to get it.

This one's very hard.


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## BJClark

smhbbag;




> Protractors are cheating



Using any of the hints would be cheating..but she looked at the hint and took advantage..


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## VictorBravo

Thanks for posting this. It was remarkably challenging (if you agree to use only elementary geometry).

But the solution is easy, once it jumps out at you. It took me all evening.


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## Zenas

smhbbag said:


> World's Hardest Easy Geometry Problem
> 
> The directions are there, but I'll post them:
> 
> 
> 
> 
> Using only elementary geometry, determine angle x. Provide a step-by-step proof.
> 
> You may only use elementary geometry, such as the fact that the angles of a triangle add up to 180 degrees and the basic congruent triangle rules (side-angle-side, etc.). You may not use more advanced trigonomery, such as the law of sines, the law of cosines, etc. There is a review of elementary geometry below.
> 
> This is the hardest problem I have ever seen that is, in a sense, easy. It really can be done using only elementary geometry. This is not a trick question.
> 
> 
> 
> 
> The figure is at the link.
> 
> Have fun.
> 
> At first glance, my first thought was "that's trivial." But it's not. Many times I thought I had quick solutions, and I did not. Yall enjoy, and without wasting much more of my own time, I'm simply going to trust the mathematicians who have said elsewhere that it is genuinely hard.
> 
> I feel like a sham of a math tutor, now.
Click to expand...


I have a few questions before I begin to answer the question:

1. What is "x"?

2. What is an angle?

3. What is a proof?

4. What is geometry? 

-Thanks.


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## smhbbag

> Thanks for posting this. It was remarkably challenging (if you agree to use only elementary geometry).
> 
> But the solution is easy, once it jumps out at you. It took me all evening.



What was your solution? No offense, but I'm willing to bet you're wrong. If not on the solution, then probably on the proof 

The full proof and answer is here 

相忘于江湖


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## VictorBravo

smhbbag said:


> Thanks for posting this. It was remarkably challenging (if you agree to use only elementary geometry).
> 
> But the solution is easy, once it jumps out at you. It took me all evening.
> 
> 
> 
> 
> What was your solution? No offense, but I'm willing to bet you're wrong. If not on the solution, then probably on the proof
> 
> The full proof and answer is here
> 
> 相忘于江湖
Click to expand...


Heh. I don't have a way right now to post my scatchings that are on a scrap of paper, but my solution was similar to his. I followed a slightly different path, but applied the same concepts. Certainly identifying the various isosceles triangles and bisecting angles with perpendiculars was key.

Instead of an equilateral triangle in the middle, I developed a mirror image of the triangle containing the angle x that I called EDG. G is a point on the perpendicular bisecting angle CDB. And then I drew a line from point D to what I called point H that was parallel to line AB. 

From that, it was a matter of bisecting angle ACB and then staring at all the angles to realize the realtionships.


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## smhbbag

> Heh. I don't have a way right now to post my scatchings that are on a scrap of paper, but my solution was similar to his. I followed a slightly different path, but applied the same concepts. Certainly identifying the various isosceles triangles and bisecting angles with perpendiculars was key.
> 
> Instead of an equilateral triangle in the middle, I developed a mirror image of the triangle containing the angle x that I called EDG. G is a point on the perpendicular bisecting angle CDB. And then I drew a line from point D to what I called point H that was parallel to line AB.
> 
> From that, it was a matter of bisecting angle ACB and then staring at all the angles to realize the realtionships.



Well, then I humbly retract my skepticism, and award you the amount of 1 Thanks for your effort  I was unable to get it after a little less than an hour, and looked up the solution. Bravo.


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## Zenas

smhbbag said:


> Bravo.



Victor Bravo.


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## smhbbag

Nice.


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## VictorBravo

smhbbag said:


> Well, then I humbly retract my skepticism, and award you the amount of 1 Thanks for your effort  I was unable to get it after a little less than an hour, and looked up the solution. Bravo.




Well, it took me a lot longer than an hour. I started thinking about it last night and probably drew triangles and lines in my head for maybe an hour last night before I fell asleep. Then I fiddled with it this evening for about 3 hours. It is a doozy.


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## VictorBravo

I don't know why, but I love these kinds of problems.

For what it's worth, I finally got my proof cleaned up and into a web suitable form. It is quite similar to the answer already posted, but I think the graphics help explain it a bit.

















So GF equals DF which equals EF. Triangle DEF is isosceles. This means that angle DEF equals angle EDF.

Angle DFE equals 80 because DF is parallel to AB. Angle DEF is found by subtracting 80 from 180 and dividing by 2, which equals 50.

Simple addition tells us that AEB is 30 degrees. Angle x is therefore 50 - 30 = 20


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## Notthemama1984

I thought I was going to get this problem eventually, but I was wrong. I guess I don't remember my geometry as well as I use to.


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## smhbbag

> I thought I was going to get this problem eventually, but I was wrong. I guess I don't remember my geometry as well as I use to.



I think it's a lot more about creativity than knowing the rules. So, it's not really a knock on our memories, just our problem-solving skills


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## Notthemama1984

smhbbag said:


> I thought I was going to get this problem eventually, but I was wrong. I guess I don't remember my geometry as well as I use to.
> 
> 
> 
> 
> I think it's a lot more about creativity than knowing the rules. So, it's not really a knock on our memories, just our problem-solving skills
Click to expand...


Oh if that is the case.......


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