What is the word for the movement you make when you give someone your hand?

Solution 1:

"Extending your hand" is indeed correct, understood exactly in this way, and of long use for precisely this. At least one dictionary even lists "extend your hand" as a phrase with precisely that meaning. The OED doesn't, but does have a definition:

To stretch forth (the arm or hand). Also, to hold out, put forward (a staff, etc.).

And includes examples of Shakespeare and Dryden using it in this way.

All in all, you couldn't ask for a more apt word choice.

Solution 2:

Both extend and offer are correct and current usage.

• "As we were walking I offered her my hand."

• "I offered my hand and he shook it."

• "I approached him and extended my hand."

• "He stared at me for a moment and then reluctantly extended his hand."

Solution 3:

"Hold out a hand" is one good option I don't see mentioned here. It's more casual and natural than extend or offer.

Here's an example of the exact situation you described:

Renly held out his hand to help Margaery to her feet.

But it can also be used when offering a hand to be shaken:

Alfred held out his hand. Jack hesitated momentarily, then shook it.

This last use is more frequent, but that's just because people shake hands more often than they help each other up.

Solution 4:

Reach out is a useful expression:

• Lit. to extend one's grasp outward.

  • He reached out, but there was no one to take hold of. I reached out and grabbed onto the first thing I could get hold of.