The value of x and how?
Q. HARDEST EASY MATH PROBLEM
Solve for x

Hi student,
Please see the diagram for the problem below ,
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Now see the step wise solution 
1. Calculate some known angles:

• ACB = 180-(50+30)-(60+20) = 20°
• AEB = 180-60-(50+30) = 40°

2. Draw a line from point E parallel to AB, labeling the intersection with AC as a new point F and conclude:

• FCE  ACB
• CEF = CBA = 50+30 = 80°
• FEB = 180-80 = 100°
• AEF = 100-40 = 60°
• CFE = CAB = 60+20 = 80°
• EFA = 180-80 = 100°

3. Draw a line FB labeling the intersection with AE as a new point G and conclude:

• AFE  BEF
• AFB = BEA = 40°
• BFE = AEF = 60°
• FGE = 180-60-60 = 60° = AGB.
• ABG = 180-60-60 = 60°

4. Draw a line DG. Since AD=AB (leg of isosceles) and AG=AB (leg of equilateral), conclude:

• AD = AG.
• DAG is isosceles
• ADG = AGD = (180-20)/2 = 80°

5. Since DGF = 180-80-60 = 40°, conclude:

• FDG (with two 40° angles) is isosceles, so DF = DG

6. With EF = EG (legs of equilateral) and DE = DE (same line segment) conclude:

• DEF  DEG by side-side-side rule
• DEF = DEG = x
• FEG = 60 = x+x

Answer: x = 30°

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Thanks & Regards ,
Harshit Tyagi