A similarity between optical and geometrical isomerisms is that

both are included in stereoisomerism.

The correct answer is option 3. both are included in stereoisomerism.

Let us explore in detail the similarity between optical and geometrical isomerisms:

Optical Isomerism:

Optical isomerism, also known as chiral isomerism, occurs in molecules that are chiral. A molecule is chiral if it lacks an internal plane of symmetry and has at least one chiral center (an atom usually carbon bonded to four different substituents). Chiral molecules exist as pairs of non-superimposable mirror images called enantiomers. Enantiomers rotate plane-polarized light in equal but opposite directions and exhibit optical activity.

Example: Lactic acid \((CH_3CHOHCOOH)\) has a chiral center at the central carbon atom due to different substituents attached to it \((-CH_3, -OH, -COOH, -H)\). It exists as two enantiomers: (+)-lactic acid and (-)-lactic acid.

Geometrical Isomerism:

Geometrical isomerism, also known as cis-trans isomerism, arises due to the restricted rotation around a double bond or ring structure. Cis-trans isomers have different spatial arrangements of atoms or groups across the double bond or ring, leading to different physical properties but the same chemical formula. Example: 1,2-dichloroethene \((C_2H_2Cl_2)\) exists as cis- and trans-isomers. In the cis-isomer, the two chlorine atoms are on the same side of the double bond, whereas in the trans-isomer, they are on opposite sides.

Similarity Between Optical and Geometrical Isomerisms:

Inclusion in Stereoisomerism:

The main similarity between optical and geometrical isomerisms is that both are types of stereoisomerism. Stereoisomers are compounds that have the same molecular formula and connectivity of atoms but differ in the spatial arrangement of these atoms or groups. Optical isomerism and geometrical isomerism represent different ways in which stereoisomers can manifest:

Optical isomerism is based on the presence of chiral centers and the resulting asymmetry in molecular structure. Geometrical isomerism arises due to the restricted rotation around double bonds or in cyclic structures.

Incorrect Options:

Option (1) each forms an equal number of isomers for a given compound is incorrect because the number of optical and geometrical isomers can vary widely depending on the compound's structure.

Option (2) if one is present, then so is the other is incorrect because a compound can exhibit optical isomerism without necessarily having geometrical isomerism, and vice versa.

Option (4) they have no similarity is incorrect because both optical and geometrical isomerisms involve stereoisomerism, albeit through different mechanisms.

Conclusion: The correct answer, (3) both are included in stereoisomerism, highlights that both optical and geometrical isomerisms are types of stereoisomerism, differing in their mechanisms but sharing the characteristic of varying spatial arrangements while retaining the same molecular connectivity. Understanding these distinctions is fundamental in organic chemistry for predicting and explaining the properties and behaviors of different types of isomers.