我从窗户向他挥手，但他没有注意到。

I ___waved at him from the window, but he didn't notice.

她非常生气，甚至不愿意向我们挥手道别。

She was so upset that she wouldn't even ___wave goodbye to us.

她向我挥手，我报以微笑。

She ___waved at me and I smiled back.