How to connect structures in Avogadro at certain angles?
Did you try a (Universal Force-Field) geometry optimization? I would: 


Select all atoms
Deselect (hold Ctrl + click) the 4 (6? 8?) carbons of interest
Click Extensions -> Molecular Mechanics -> Ignore Selection 
Click Extensions -> Optimize Geometry


Alternatively, you could optimize a few more atoms around/in the bridge while the others are constrained. The G.O. algorithm is pretty robust and handles many systems quite well. A simple example of hand-drawn cyclohexane:


I got a result like this by making 2 nanotubes with chiral index (4,4), drawing 6 bonds and optimizing the bridge:




If this is ineffective, I would resort to:


Change the 4 carbons to different elements e.g.


$\ce{O        H}$

$\ce{N        S}$

(like the arrangement you have in the picture, so you can identify them later)


Click File -> Save As... and save the file as an .xyz
Calculate the O-N and H-S distance by $r_{ij} = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$ from the coordinates in the .xyz file.
Ideally, Avogadro saved the .xyz coordinates aligned in z (which is default when you draw molecules without rotating the initial view).
(Even if not,) re-write the coordinates of the atoms aligned in x and z, e.g. $\ce{O}$ at $(x, y, z)$; $\ce{N}$ at $(x, y-r, z)$.
Or (yikes) write a quick program to generate possible systems with the fixed $r$ you calculated, name them system-000.xyz, system-001.xyz, system-002.xyz, etc., and view the "trajectory" frames until you find one pleasing. (Viewing can be made much easier with VMD, e.g. vmd system-*.xyz).




Avoiding Avogadro: If you use Molden and load your system (as an .xyz, .car, .pdb etc.), you can use the ZMAT-editor to manually change bond angles and dihedrals:





Finally, if all else fails, it may be time to consider a full quantum-mechanical treatment of the system (if you're familiar with High Performance Computing) with a free (for academia!) program like Orca (where one can constrain certain atoms during the optimization). An example input would be: 

! hf 6-31g* opt  # or copt, for cartesian optimization

%pal
nprocs 128   # num. processors
end

%scf
maxiter 2000 # max self-consistent field iterations
end

%geom
Constraints # atoms to keep fixed starting at index 0
{C 61 C}
{C 62 C}
{C 290 C}
{C 291 C}
{C 292 C}
end
end


*xyz 0 1  # charge=0; mult=2S+1 = 1 (closed shell)
  C      -1.306743      4.170005      1.012072
  C      -0.042414      3.718873      0.686759
  C       0.210647      2.367809      0.785537
  C      -2.267754      3.271952      1.435683
  C      -2.920909      0.889649      1.948837
  C      -4.153277      1.187877      2.498723
  C      -4.989183      0.155126      2.896328
  C      -4.571713     -1.152339      2.735382
  C      -3.332221     -1.380404      2.172864
....... etc.
*