[Example 4.4.4] There is a tetrahedron $A-BCD$, where the sum of the dihedral angles at each vertex is $180^{\circ}$, and it has three unequal edge lengths of $\sqrt{34}$, $\sqrt{58}$, and $\sqrt{74}$. Find the volume of this tetrahedron.
We first study the characteristics of this tetrahedron. Unfold the tetrahedron, i.e., place $\triangle A B C, \triangle A C D, \triangle A B D$ on the plane of $\triangle B C D$.

Since $\angle A B D+\angle C B D+\angle A B C=180^{\circ}$, then $A_{1}, B, A_{2}$ are collinear, similarly $A_{2}, C, A_{3}$ are collinear, and $A_{1}, D, A_{3}$ are collinear.

At this point, $\triangle A_{1} B D, \triangle B A_{2} C, \triangle D C A_{3}$, and $\triangle C D B$ are congruent triangles, thus $A_{1} B=A_{2} B$ $=C D, A_{1} D=A_{3} D=B C, A_{2} C=A_{3} C=$ $B D$, meaning that the pairs of opposite edges of the tetrahedron $A-B C D$ are all equal. Therefore, the three unequal edge lengths given in the problem are the lengths of the three pairs of opposite edges.

For a tetrahedron with equal opposite edges, we can complete it into a rectangular prism $M D N C-B P A Q$, where the face diagonals $A B$ and $C D, A D$ and $B C, A C$ and $B D$ are the three pairs of opposite edges of the tetrahedron $A-B C D$. At this time, the volume $V$ of the tetrahedron can be found through the volume of the rectangular prism.
Let the length, width, and height of the rectangular prism be $a, b, c$, respectively, then we have
$$
\left\{\begin{array}{l}
a^{2}+b^{2}=34 \\
b^{2}+c^{2}=58 \\
c^{2}+a^{2}=74
\end{array}\right.
$$

Solving this, we get
$$
a=5, b=3, c=7 .
$$

Thus,
$$
V=V_{\text {MDNC-BPAQ }}-4 V_{B-M C D} \text {, }
$$

which is
$$
V=a b c-4 \cdot \frac{1}{3} \cdot \frac{1}{2} a b c=\frac{1}{3} a b c=35 .
$$

Completing the tetrahedron with equal opposite edges into a rectangular prism is a crucial step in solving this problem.
A tetrahedron with equal opposite edges is called an isosceles tetrahedron. Completing an isosceles tetrahedron into a rectangular prism with length, width, and height $a, b, c$ not only allows us to find the volume of this isosceles tetrahedron but also reveals many properties, such as:
The faces of an isosceles tetrahedron are all acute triangles.
The lines connecting the midpoints of the opposite edges of an isosceles tetrahedron intersect at one point and are perpendicular and bisect each other at this point. The lengths of the lines connecting the midpoints of the opposite edges of an isosceles tetrahedron are exactly the length, width, and height of the rectangular prism.