What is the value of $\angle x$

Question

Can any one help with this problem:what is value of $\angle x$
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Did you take into account the fact that you have rectangular triangles and that the sums of angles in a triangle is Pi ? — Claude Leibovici, Jan 03 '14 at 06:04

mathlove · Answer 1 · 2014-01-03T07:15:14.773

The answer is $30^\circ$.

Proof: Let this big triangle be $ABC$ where $$\angle ABC=20^\circ+60^\circ, \angle ACB=30^\circ+50^\circ, \angle BAC=20^\circ.$$

Also, let $D,E$ be the inner point on $AC, AB$ respectively.

Now, take a point $F$ on $AB$ such that $BD=BF$. Also, take a point $G$ on $DF$ such that $AD=AG.$ (Note that $G$ is outside of $ABC$)

Also, prove that the triangles $AGF, DBC$ are congruent. Then finally prove that $FD=FE$.

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