(Polar molecules, Non-polar molecules, etc.)
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I'm pretty sure that to determine whether or not atoms are in the same plane you have to see if there is the presence of a pi bond, which results from either a double or triple bond between atoms. Otherwise, if there is just a single bond/sigma bond, the atoms are free to rotate.
Even if the molecule is "bent" or "angular" the molecule as a whole should be found in a single plane. This is true for those "bent" shaped molecules, but also for molecules that are trigonal "planar" and square "planar" (the name suggests that they lie in such a plane).
Atoms will be in the same plane if they have a double or triple bond since they can't rotate due to having a pi bond. Furthermore, if they are either trigonal or square planar the atoms also lie in the same plane.
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