SSL is half symmetric and half asymmetric?

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I am reading http://www.definityhealth.com/marketing/how_ssl_works.html

Looks like SSL is using asymmetric algorithm to exchange the symmetric key, after that it uses symmetric algorithm to encrypt the data.

One question, can I use asymmetric algorithm only? Like Alice and Bob both have certificate and, they are all using peer's public key to encrypt the data.

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