It is known that $a,b,c>0$ are the sides of a triangle in the Euclidean plane if and only if $$a+b>c,\hspace{0.3cm} a+c>b,\hspace{0.3cm} b+c>a.$$ I would like to give a similar condition for spherical triangles (geodesic triangles in $S^2\subset \mathbb{R}^3$).
In the Euclidean case you just pick a segment of side $a$ and then you consider the intersection points between two circles. One circle with center the left edge of the segment and radius $b$ and the other with center the right edge and radius $c$. If you choose coordinates and you make the computations you can prove that this intersection is not empty if and only if the conditions above are satisfied. Also, if the intersection is not empty, it has two points symmetric with respect to our initial segment.
I expect that something similar happens in the spherical case. This has been my attemp so far:
We can assume without loss of generality that the situation is the following. Our triangle has vertices $$A=(\cos(\varphi)\cos(\theta), \sin(\varphi)\cos(\theta),\sin(\theta)), \hspace{0.2cm} B=(1,0,0), \hspace{0.2cm} C=(\cos(a),\sin(a),0), $$ where $\varphi \in(0,2\pi)$ and $\theta \in(-\pi/2,\pi/2)$. It is known that for any two points $P,Q\in S^2$ we have that $\cos(\overline{PQ})=\langle A, B \rangle$. Then, we can choose sides $a,b,c>0$ satisfying $\cos(\overline{AB})=\cos(c)$, $\cos(\overline{AC})=\cos(b)$, $\cos(\overline{BC})=\cos(a)$. This implies that the following holds \begin{align} \cos(c)&=\cos(\varphi)\cos(\theta)\\ \cos(b)&=\cos(a)\cos(\varphi)\cos(\theta)+\sin(a)\sin(\varphi)\cos(\theta). \end{align} If we impose that $0\leq a\leq b\leq c\leq \pi/2,$ which implies that the triangle is contained in an octant I think that this system has two solutions as in the Euclidean case.
But I can't solve it and I can't see which are the conditions over $a,b,c$ for this system to have a solution.
Hint: Start by reading section 18.6 in
M.Berger, "Geometry", Volume II.
You will discover that you are missing one inequality coming from the fact that the angle sum of the polar spherical triangle is $>\pi$. Once you know the full set of inequalities, proving the existence is not difficult and is similar to Euclid's proof in the case of euclidean triangles.