## World's hardest easy geometry problem?

The tag "World's hardest easy geometry problem" was coined by Keith Enevoldsen, who provides an interesting commentary on the task, and who traces its origins back to a journal published in the 1970s. Of course, it might have existed for much longer (perhaps since the big bang if you think god might have had a hand in it).

Enevoldsen says he worked on the task "for many hours over several days" before solving it. We had a similar experience, yet the task can be solved using only basic geometry in just a few steps - which you might hit upon in a matter of minutes!

Enevoldsen presents another such task on his website: the world's second hardest easy geometry problem. Are such tasks interesting? We're not sure - they don't seem to involve any kind of interesting (general) mathematical stucture, and Enevoldsen aptly describes them as adventitious.

However, it is easy to get hooked, sufficiently so in our case to see whether we could find more such tasks. We conducted a search using Geogebra which allowed us systematically to change the given angles in the task. To keep things manageable, we restricted ourselves to angles that were multiples of 5˚. In the event we came up with 8 more examples where the required angle AED turns out to be a nice round number, which one imagines therefore could be found using simple geometry. We have managed to do this with 6 of the tasks so far (some of which turned out to be very easy) but we are still completely stuck on two of them.

The set of tasks can be found using the links at the top of the page.
"10-10-20" refers to Enevoldsen's first task (where half the angle at C is 10˚, where the 'outer' angle at A is 10˚ and the 'outer' angle at B is 20˚). 10-20-30 is Enevoldsen's second task. The last task, 10-20-45, is a rogue - the size of angle E appears to be a round number, but in fact it isn't!

Note: In our versions of these tasks, the drawing are to scale and the value of the desired angle is shown - the challenge is to prove that this value is correct.

Using only elementary geometry, determine angle x.
Provide a step-by-step proof.