My friend sent me the following geometry problems. I think I have the first one, but I think the 2nd and 3rd are unsolvable, although I could be missing something.
My attempt:
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I'm pretty sure this one is 1080. I'm having a hard time writing out my explanation here, but I can justify it on paper. The idea is that the outer shape (if we ignore the triangles) is an octagon, and the sum of its angles is 1080, and I can show that the sum of the marked angles is also 1080.
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I don't think this one gives us enough information. If we moved the point D left or right, the angle of $x$ would change and the constraints would still be satisfied.
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I can't figure out the answer to this one. I think it's unsolvable as well, but I can't prove it like #2. We can't find the area of anything in this picture. The shaded region and the whole shape are both close to being a trapezoid, but they aren't.
Best Answer
$(1)$ It looks unsolvable. Yes, breaking it into the "outer octagon" and the "inner triangles" is a good idea. But think, you can shift the dots to matching the intersection angles right? but you are missing the large angle in each triangle towards the "whole" angle (the sum of 3 components of the intersection) for the octagon.
$(2)$ moving $D$ messes with the equality $AD=BD$ as "the edges will stretch disproportionately." Use the fact that $AD=BD$!
$(3)$ at the risk of being rude, the question is a mess while at heart a good start, you're correct. Maybe adding an edge length or two, or an angle is all that is needed