you say Obrigado, she say Obrigada to you,

for example I would say ‘Obrigado’ to her not ‘Obrigada’.

Michelly to Bill “Obrigada Bill”

Bill to Michelly “Obrigado Michelly”

So you would say “Obrigado” to a man or woman, not “Obrigada”.

and a woman would say “Obrigada” to a man or woman.