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Using only elementary geometry, determine angle x.


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

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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
@user236182: You are right. – Aryabhata Oct 26 at 23:23

2 Answers 2

up vote 5 down vote accepted

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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