6.12. In a triangular pyramid $S A B C$ with vertex $S$, the lateral edges are equal, and the sum of the dihedral angles at edges $S A$ and $S C$ is $180^{\circ}$. Express the length of the lateral edge in terms of the sides $a$ and $c$ of triangle $A B C$.
6.12. Let's take a point \( A_{1} \) on the ray \( A S \) such that \( A A_{1} = 2 A S \). In the pyramid \( S A_{1} B C \), the dihedral angles at the edges \( S A_{1} \) and \( S C \) are equal, and \( S A_{1} = S C \), so \( A_{1} B = C B = a \). The triangle \( A B A_{1} \) is a right triangle because its median \( B S \) is equal to half of the side \( A A \). Therefore,

\[
A A_{1}^{2} = A_{1} B^{2} + A B^{2} = a^{2} + c^{2}, \text{ i.e., } A S = \sqrt{a^{2} + c^{2}} / 2
\]