Which of these powers is a constitutional power that the president shares with the senate?

The President and the Senate share two important powers. First, both parts of the government are able to create and approve treaties with other nations. Second, both can appoint ambassadors, judges, or high officials to their needed posts (usually, this is an appointment using the "advice and consent" of the Senate, who will then approve the selection if they feel the person is qualified).

What ultimately brought the US out of its neutrality was when the Germans wanted to stop supplies from getting to Great Britain from the US. So, they sunk the supply ships which also had hundreds of Americans on board which the Germans killed altogether (triggering their anger and causing them to join the war).