[Math] finding angle value inside this triangle

Question

I need a method to calculate the angle X in the image below, I know its value (30 degree) but how ?!!

thank you.

Answer 1

First of all, note that $EA=AB$. Take a point $G$ on $DB$ such that $\angle{GAB}=20^\circ$. Since $\angle{ABG}=\angle{AGB}=80^\circ$, one has $AB=AG$. So, since $\triangle EAG$ is an equilateral triangle, one has $AG=GE$. Since $\angle{GAD}=\angle{GDA}=40^\circ$, one has $AG=GD$. Hence, one has $GE=GD$. So, since $\angle{EGD}=40^\circ,\angle{EDG}=\frac{180^\circ-40^\circ}{2}=70^\circ$ (note that $\triangle EDG$ is an isosceles triangle), one has $$\angle{EDA}=\angle{EDG}-\angle{GDA}=70^\circ-40^\circ=30^\circ.$$