How  is the measure of angle marked as ?

Below is the sketch along with a few of the easier angles, which should require no explanation.

x + y = 120°

Applying the sine rule:

In ∆ADE, DE = AEsin(20°)/sin(40°)

In ∆ABE, EB = AEsin(20°)/sin(100°)

DE : EB = sin(100°) : sin(40°)

DE : EC = sin(100°) : sin(40°) ... (because of isosceles ∆ BCE)

In ∆CDE,

DE/sin(x) = EC/sin(y)

sin(x) : sin(y) = DE : EC

sin(x) : sin(y) = sin(100°) : sin(40°)

sin(x) : sin(y) = sin(80°) : sin(40°) ... (because sin(100°) = sin(80°))

So we have this:

sin(x) : sin(y) = sin(80°) : sin(40°) and x + y = 120°

x = 80°, y = 40°

The angle you required is x.

x = 80°

AEB = 180 –20–100 = 60º

DEC = AEB = 60º

AED = 180 – AEB = 120º = BEC

ADE = 180 – 20 – 120 = 40º

BCE = 180 – 30 – 120 = 30º

Triangle DCE

x + y + 60 = 180

x + y = 120

The answer is 80 degrees.

The other two angles in the same triangle are 60 and 40