Determine angle $x$ using only elementary geometry

Question

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.

Answer 1

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]

Answer 2

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