Scissor congruence of two general tetrahedrons having mirror symmetry

Question

Problem : If $P,\ P'\subset \mathbb{R}^3$ is two polyhedra which are congruent up to a mirror symmetry then prove that they are scissor congruent

Proof : We can cut polyhedron into a finite number of general tetrahedrons Hence this problem is reduced to case where $P,\ P$ are two general tetrahedrons But how can we finish the proof ?

Thank you in advance

Answer 1

Note that triangle has incentor so that we have 3 pair of triangles having mirror symmetry We can generalize this on general tetrahedron