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1,4-dichlorocyclohexane has two isomers, and 1,3-dichlorocyclohexane has three:

enter image description here

Looking at the substituted carbon, the reason for the isomers is the same: either the chlorine points up or it points down.

Why does IUPAC nomenclature use cis/trans for the former and R/S for the latter?

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    $\begingroup$ Those 1,4 carbons on the first one aren't chiral centers, and the images aren't stereoisomers: those two images are not non-imposable mirror images of each other, nor are there 4 different substituents on the 1 and 4 carbons.Thus R/S is not applicable here and cis/trans is. See this post for some more details. $\endgroup$ Commented Mar 31 at 14:49
  • $\begingroup$ @ToddMinehardt I updated the question, stating "isomers" instead of "stereoisomers". I am unsure whether geometrical isomers are a subset of stereoisomers (same Lewis structure, different 3D structure?), e.g. this figure. $\endgroup$
    – Karsten
    Commented Mar 31 at 15:18
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    $\begingroup$ The preferred IUPAC nomenclature doesn't use cis,trans for 1,4-dichlorocyclohexane. The preferred names are (1s,4s)-1,4-dichlorocyclohexane and (1r,4r)-1,4-dichlorocyclohexane, respectively. $\endgroup$
    – Loong
    Commented Mar 31 at 19:26
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    $\begingroup$ Related What does lowercase r-s notation mean? $\endgroup$
    – Loong
    Commented Mar 31 at 19:30

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Geometrical and optical isomerism are both forms of steoreoisomerism, which is isomerism that arises due to difference in orientation in 3D space, and they are not mutually exclusive.

As per IUPAC, the strict definition of geometrical isomerism is

Stereoisomeric olefins or cycloalkanes (or hetero-analogues) which differ in the positions of atoms (or groups) relative to a reference plane: in the cis-isomer the atoms are on the same side, in the trans-isomer they are on opposite sides.

RS Nomenclature may be used to refer to the absolute configuration of a chiral molecule.

1,4-dichlorocyclohexane does not possess a chiral carbon, and you cannot use RS to denote the absolute configuration. It can however have cis and trans forms

1,3-dichlorocyclohexane has two chiral carbons, and 3 stereoisomers, 2 optically active forms which are enantiomers, and 1 optically inactive (meso) form

enter image description here

Using 'cis' or 'trans' here is not wrong, as it does satisfy the necessary conditions, but there are two 'trans' forms.

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