what is the name of nine traditional dances, i.e. waltz?

Answer 1

If you are talking about 9 dances that constitute American Ballroom competition, here they are:
Waltz, Foxtrot, Vienneeze Waltz, Tango
Cha-cha-cha, Rumba, Bolero, East Coast Swing, Mambo
International Ballroom competition has 10 dances: Waltz, Foxtrot, Vienneeze Waltz, Tango, Quickstep
Cha-cha-cha, Rumba, Samba, Jive, Paso Doble.

Both International style and American style are competed in the US, only intrenational style everywhere else in the world. American style is also danced at social dances, this was the original purpose of branching it off from International.

Answer 2

Nine traditional folk dances: Schottische, Mazurka, Tarantella, Horah, Oberek, Polka, Troika, Morris Dance, Farandole.

Nine more well-known traditional dances: Fox Trot, Rhumba, Tango, Bolero, Cha-Cha, Salsa, Paso Doble, Quickstep, Samba.