extend a/(one's) hand

1. To reach toward someone in order to shake hands. He extended his hand once we agreed on the terms, and I shook it firmly. I wasn't trying to snub him, I honestly just didn't see him extend his hand to me! After Mary introduced us, I extended a hand to Tom.

2. To make a gesture or declaration of welcome, amity, goodwill, etc. (to someone). Ultimately, we are all football fans, so I always extend a hand in friendship to my opponents and their supporters. The conservatives in Congress eventually extended a hand to Democrats on the issue of raising the debt ceiling. Of course I would extend my hand to Ellen even though I don't know her personally. She's my best friend's friend, after all!

3. To offer one's support or assistance (to someone). The entire community has extended their hand to help us recover from this devastating tragedy. The country was quick to extend a hand when its neighbor to the North needed an ally. Thankfully, the government extended a hand to the hurricane-ravaged area immediately after the storm dissipated.