Then I asked her, ‘Whose daughter are you?’ She said, ‘The daughter of Bethuel, Nahor's son, whom Milcah bore to him.’ So I put the ring on her nose and the bracelets on her arms.
Genesis 24:47 ESV

Eliezer relates these events in reverse order. He actually gave her the rings and then asked her whose daughter she was. He led with the fact that would have carried less weight with his audience (blood relations), but perhaps he wanted them to put more weight on, knowing that they would be more interested in the gold.