Theoretically the angles all “work,” but I decided to actually try to make the dang thing. I started with doing the basic angle sizes from the information in the example and basic triangle/line math:
To draw the shape I started with with line AB. Then I drew guidelines out at the correct angles based on whatever X I was using (5/10/20) from A & B. C & D were found where those guidelines met. Then I connected C & D. You can see from the pictures that doing it this way results in the correct expected angles in all of the places except the CD angles.
The bottom C angle is always correct but the top one never equals X. It is too big if my chosen X is less than the given correct answer of 15, or too small the other way. This correlates in some way with what @fonserbc was trying to say earlier I think.
Maybe somebody can see this and explain why.