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Finding an angle within an 80-80-20 isosceles triangle

There's a decades-old geometry problem that's part of math folklore and that some of you have probably seen. It's called the "world's hardest easy geometry problem" at one website:

http://thinkzone.wlonk.com/MathFun/Triangle.htm

In order to consider generalizations, consider the following picture, where the large triangle ABC is isosceles with angles 80°, 80°, 20°.

triangle

Problem 1: Given $\alpha=70^\circ$ and $\beta=60^\circ$, find $\theta$.

Problem 2: Given $\alpha=60^\circ$ and $\beta=50^\circ$, find $\theta$.

The challenge is to find elementary solutions to these problems, which in this context means no trigonometry.

I have already found (with some help) elementary solutions to those two specific problems. The elementary solution to Problem 1 is different from the elementary solution to Problem 2. Both solutions are very specific, and involve drawing extra lines and then showing certain lengths are equal and certain angles are equal.

I'm not primarily interested in Problems 1 and 2 themselves. I'm interested in a more general (although perhaps not completely well-defined) question.

If trigonometry is allowed, one can find a general expression for $\theta$ in terms of $\alpha$ and $\beta$. However, that expression is very unwieldy for hand computation. But with the help of a computer, we can find $\theta$ numerically when given specific values of $\alpha$ and $\beta$.

For some "round" values of $\alpha$ and $\beta$, we numerically find that $\theta$ appears to be a "round" value as well. The following five statements are accurate to at least eight significant digits:

(i) If $\alpha=70^\circ$ and $\beta=50^\circ$, then $\theta=10^\circ$.

(ii) If $\alpha=60^\circ$ and $\beta=30^\circ$, then $\theta=10^\circ$.

(iii) If $\alpha=50^\circ$ and $\beta=40^\circ$, then $\theta=30^\circ$.

(iv) If $\alpha=50^\circ$ and $\beta=20^\circ$, then $\theta=10^\circ$.

(v) If $\alpha=65^\circ$ and $\beta=60^\circ$, then $\theta=40^\circ$.

My two questions (in admittedly slightly vague form) are (A) Are these five statements exactly true? and (B) Why?

In slightly more precise form:

(A) Can we prove that statements (i)--(v) are exactly true and not just numerically true? Can we do this algebraically, or geometrically? Can we prove them by any kind of general argument, or do we have to use different arguments for the different statements?

(B) For brevity, call an angle "pi-rational" if it is a rational multiple of $\pi$ radians or 180 degrees. Can we characterize, or partly characterize, the pi-rational values of $\alpha$ and $\beta$ that will result in $\theta$ being pi-rational?

By the way, note that some particular instances of the problem appear at the following webpage

http://www.cut-the-knot.org/triangle/80-80-20/index.shtml

but only some of them have solutions there.

Also, I should probably give my formula for $\theta$ in terms of $\alpha$ and $\beta$. I can explain where this came from if needed. We have $ \theta = \arcsin\Big(\frac{\sin(\alpha+\beta)}{\sqrt{1+\rho^2+2\rho\cos(\alpha+\beta)}}\Big) $ where $ \rho = \frac{\sin(\alpha)\sin(80^\circ-\beta)\sin(100^\circ-\beta)}{\sin(\beta)\sin(80^\circ-\alpha)\sin(100^\circ-\alpha)} $

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    Re the appearance of $80^\circ$ and $100^\circ$: yes, that's exactly why. There's no deep reason I didn't consider $\gamma$ to be variable, beyond the fact that the problem is already quite tricky with a specific $\gamma$. And thank you for the terms "adventitious angles" and "Langley's problem". This helps immensely.2011-12-30

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