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Using only elementary geometry, determine angle x.

triangle

You may not use trigonometry, such as sines and cosines, the law of sines, the law of cosines, etc.

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1  
Sheesh, and I thought the law of cosines was elementary... –  J. M. Sep 12 '11 at 5:03
    
The world's hardest easy geometry problem... :) –  tom Sep 12 '11 at 13:21
    
possible duplicate of Finding an angle within an 80-80-20 isosceles triangle –  Aryabhata Sep 15 '11 at 2:26

2 Answers 2

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adventitious angles

  1. Draw a line DF parallel to AB, intersect BC at F;
  2. Connect AF, intersect BD at G;
  3. Connect CG.

Now, it's easy to prove that CE=AG, and DF=DG=GF. Since AF=CF, then EF=GF.

Then EF=DF $\Rightarrow$ $\angle$FED=$\angle$FDE.

While $\angle$DFE=$\angle$ABC=80$^\circ$, so $\angle$DEF=50$^\circ$.

From $\angle$AEB=30$^\circ$, we can get x=$\angle$DEA=20$^\circ$. [Q.E.D]

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This is known as the problem of "adventitious angles". You'll find many references if you search the web for that phrase.

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Wow, so that is the name for those things... thanks! I'm now looking at this... –  J. M. Sep 12 '11 at 5:58

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