Why is halogenation anti addition?

........when a halogen adds to a cyclic olefin, it forms a so-called bromonium ion, a three-membered ring......

The picture depicts the supposed intermediate after the bromine adds (as an electrophile) to the olefin. We might now invoke a carbocation intermediate, which of course will undergo subsequent reaction with the newly delivered bromide nucleophile....

#R_2C=CH_2 + Br_2 rarr R_2BrC-stackrel(+)CR_2 +Br^-#

The formal carbocation MAY be stabilized somewhat by the formation of a bromonium ion, in which the added bromine atom forms a three-membered ring with the potential or conceptual carbocation, thereby somewhat stabilizing it. The newly delivered bromide ion is DIRECTED AWAY from that side of the ring, and thus we get a trans-disubstituted alicycle.

Note that this stereochemical peculiarity is also observed when #Br_2# is added to #"cis-2-butene"# versus #"trans-2-butene"#.

Because of the anti-addition, #"cis-2-butene"# gives the #R,R# and #S,S# enantiomers, whereas addition to #"trans-2-butene"# gives the optically inactive #R,S#, and #S,R# #"meso"# compound. This simple experiment is good evidence for the intermediacy of a three-membered ring, which blocks off ONE side of the molecule, and results in the given stereochemistry.

Anyway, look in your text, because this is an excellent example of a so-called #"stereospecific reaction"# in which the mechanism of the reaction determines the stereochemical outcome of the reaction.