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Let's say I have a tetrahedron like this in image:

enter image description here

Do angles $CAD$ and $CBD$ equals in general tetrahedron?

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Say you have a triangle $BCD$ in three-dimensional space. Then the angle $\angle CBD$ is already decided, while you can put the point $A$ wherever you want, and get whatever angle you like for $\angle CAD$, within a given interval (likely $(0, \pi)$, if you like your angles measured internally and your tetrahedron non-degenerate). So the answer is no.

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