10.6  Let the insphere of a pyramid $SABC$ touch the faces $SAB, SBC, SCA$ at $D, E, F$ respectively. Find all the possible values of the sum of the angles $SDA, SEB, SFC$.
1. Let the insphere of the pyramid \( SABC \) touch the faces \( SAB, SBC, SCA \) at points \( D, E, F \) respectively. Let \( G \) be the contact point of the insphere with the base face \( ABC \).

2. Notice that the triangles \( \triangle SDA \) and \( \triangle SFA \) are congruent because they have equal corresponding sides. Similarly, the triangles \( \triangle AFC \) and \( \triangle AGC \) are congruent, so \( \angle AFC = \angle AGC \).

3. By similar reasoning, we can show that:
   \[
   \angle AFC + \angle CEB + \angle BDA = \angle AGC + \angle CGB + \angle BGA = 360^\circ
   \]

4. Therefore, we have:
   \[
   \angle SDA + \angle SFC = 360^\circ - \angle AFC
   \]

5. Adding the cyclically obtained relations, we get:
   \[
   2(\angle SDA + \angle SEB + \angle SFC) = 1080^\circ - (\angle AFC + \angle CEB + \angle BDA) = 720^\circ
   \]

6. Dividing both sides by 2, we find:
   \[
   \angle SDA + \angle SEB + \angle SFC = 360^\circ
   \]

The final answer is \(\boxed{360^\circ}\).