116. In a triangular pyramid $S A B C$ with base $A B C$ and equal lateral edges, the sum of the dihedral angles
with edges $S A$ and $S C$ is $180^{\circ}$. It is known that $|A B|=a_{s}$ $|B C|=b$. Find the length of the lateral edge.
116. Extend the edge $S A$ beyond point $S$ and take a point $A_{i}$ on the extension such that $\left|S A_{1}\right|=|S A|$. In $S A_{1} B C$, the dihedral angles at the edges $S A_{i}$ and $S C$ will be equal, and since $\left|S A_{1}\right|=|S C|$, then $\left|A_{1} B\right|=|C B|=b$. Triangle $A B A_{1}$ is a right triangle with legs $a$ and $b$. Therefore, the hypotenuse $\left|A A_{1}\right|=2|A S|=$ $=\sqrt{a^{2}+b^{2}}$.

Answer: $\frac{1}{2} \sqrt{a^{2}+b^{2}}$.