Is it true that if a triangle on a unit sphere has 2 sides with equal length then their opposit angles must be equal? I think it is true. I think we can use the spherical sine law. Call the sides with equal lengths $a,b$ and their opposite angles $\alpha,\beta$. Then since $a=b$, $\sin\alpha=\sin\beta$. How do I then say for certain that $\alpha=\beta$? I know that the angles must be $\in (0,\pi)$ (right?). But how can I exclude the possibility of one angle being $\pi-$ the other angle?
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(bit late)... But since you were working on using the sine rule, you can say that $$\sin(\alpha )=\sin(\beta)\implies\alpha=\beta$$
The third angle is $0$, because the sum of the three is equal to $\pi$.