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I've been trying stuff with angle sums, Sine law and Thales theorem, but either I'm making very bad mistakes or I'm just tired - either way I would love some outside input. Thank you.

triangle

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  • $\begingroup$ This is essentially a duplicate of "Finding angle in isosceles triangle", although a different angle is sought. The sole answer uses no trig. (I seem to recall another duplicate, but I haven't been able to locate it.) $\endgroup$ – Blue Nov 8 '18 at 10:23
  • $\begingroup$ Ah, here's another: "Find an angle of an isosceles triangle ", with a variety of answers. There could be more. $\endgroup$ – Blue Nov 8 '18 at 10:26
  • $\begingroup$ Thanks a lot for the help, I appreciate it. Btw, when you searched for these questions did you just used keywords from my question or is there some cleverer way of searching on here? $\endgroup$ – Rock Dekasba Nov 8 '18 at 11:48
  • $\begingroup$ Searching Math.SE can be tricky. The task is made that much harder when question titles are really generic, instead of describing what the question is actually about. (hint, hint) ;) In this case, I just searched for something like "isosceles angle 20". (Thinking I might've answered the question myself at some point, I tried a search with "user:409" as well.) Since I was pretty sure I'd seen the question before, I probably sifted-through a few more generically-titled questions than a casual searcher might have done. $\endgroup$ – Blue Nov 8 '18 at 12:02
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I'll leave out the degree symbol from my equations.

Let'd denote $\gamma$ as the angle BDA. Then the angle BDC $=180 - \gamma$.

Let's also denote $\beta$ as the angle DBC. From symmetry, we get that the angle ACB $=\alpha + \beta$. Now, looking at the lower triangle, we obtain $$ (\beta) + (\alpha + \beta) + (180-\gamma) = 180 \qquad \Rightarrow \qquad \alpha + 2\beta - \gamma = 0 $$ From the upper triangle we get $$ 20 + \alpha + \gamma = 180 \qquad \Rightarrow \qquad \alpha + \gamma = 160 $$ Maybe you can continue from here?

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  • $\begingroup$ thank you for the reply - I'm still having trouble continuing from there, I had these angle sums before $\endgroup$ – Rock Dekasba Nov 8 '18 at 9:03
  • $\begingroup$ You can additionally use the en.wikipedia.org/wiki/Law_of_sines for $\alpha$ and the $20$ degree angle. $\endgroup$ – Matti P. Nov 8 '18 at 9:06
  • $\begingroup$ That's what I've been doing - I've written down the sine law for the 3 triangles, I know that AB=AC and AD=BC so I assume it has something to do with that but I've been at this for hours and I'm either too dumb to figure out what I'm supposed to be comparing or just doing it wrong, this is my first time doing gemotery in about 10 years so I'm sorry if I'm asking to be spoonfed or something $\endgroup$ – Rock Dekasba Nov 8 '18 at 9:31
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    $\begingroup$ @RockDekasba: You seem to have done some thinking about the problem. You should post your thoughts as part of the question itself, so that people don't waste time duplicating your effort and/or telling you things you already know. This also helps answerers tailor their responses to your experience level. (Plus, it helps convince people that you aren't just trying to get them to do your homework for you. :) $\endgroup$ – Blue Nov 8 '18 at 10:33

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