Today, as Nick Schifrin reports, the U.S. and more than a half-a-dozen other countries recognized Juan Guaido, currently the head of Venezuela's National Assembly, as the country's legitimate president.

On a stage in downtown Caracas, in front of a crowd of thousands, 35-year-old Juan Guaido raised his right hand and administered his own oath of office.

I swear to formally assume the powers of the national executive as the president in charge of Venezuela.

Around the capital and country today, hundreds of thousands, perhaps millions of Venezuelans rallied in support of Guaido and called for a change in government.

That is why we are here, to support our National Assembly, the only legitimate power for the 14 million Venezuelans.

Nuns demanded freedom, and other protesters demanded President Nicolas Maduro step down.

Your time is up, and your Cabinet's.

Understand this: Venezuela has outgrown you.

Venezuelans have protested before, but this time the usually fractured opposition has a consensus leader.

Until recently, Guaido was relatively unknown.

But he has crisscrossed the country speaking against Maduro, asking for support from the international community, and Venezuela's powerful military.

We are not asking you to mount a coup, or to shoot.

On the contrary, we are asking you not to shoot at us, and defend together with us the right of our people to be heard.

His calls have been heard.

On Monday, National Guardsmen posted cell phone videos declaring Maduro illegitimate and calling for protests.

MIKE PENCE, Vice President of the United States: We say to all the good people of Venezuela, estamos con ustedes.

Yesterday, Vice President Pence promised the U.S. would be with the protesters.

And today, in a statement, President Trump endorsed Guaido as interim president and said: "The people of Venezuela have courageously spoken out against Maduro and his regime and demanded freedom and the rule of law."

MOISES RENDON, Center for Strategic and International Studies: This is one of the most historic days for Venezuelan modern history.

Moises Rendon is an associate director of the Center for Strategic and International Studies.

He says Guaido's announcement today, and President Trump's endorsement, allows the U.S. to redirect payments for Venezuelan oil, which accounts for 90 percent of Venezuela's GDP.

Every single asset, every single bank account, every single contract, all the management of this republic's assets will be transferred to the National Assembly and to Juan Guaido.

But Maduro is pushing back.

Today, he cut off diplomatic relations with the U.S., and gave U.S. diplomats 72 hours to leave the country.

NICOLAS MADURO, Venezuelan President: Do not trust the gringo empire.

The gringos do not have friends, nor are they loyal to anyone.

The gringos have interests, Venezuelan oil, Venezuelan gas, Venezuelan gold.

But, to that empire, we say that oil, that gas, that gold is not yours.

It was just two weeks ago Maduro was inaugurated and swore to build what he called 21st century socialism.

But he has built an economic catastrophe.

Venezuela used to be Latin America's wealthiest country.

Caracas residents have demanded access to a supermarket, even if the shelves were empty because of a shortage of food.

Medical patients have protested a shortage of medicine.

And children play in the dark because of a shortage of power.

It's been an economic freefall, the product of falling oil prices and failed economic policies.

Bills have became so worthless, women turned them into art.

Inflation could hit 10 million percent.

All of it sparked the region's largest ever exodus.

More than 3.5 million Venezuelans have fled their homes and created a humanitarian crisis that increased regional criminality.

It represents a humanitarian and a security crisis; the region that is impacting not only neighboring countries, but like the U.S. as well.

The U.S. has imposed sanctions on Maduro and his leadership.

U.S. officials say, depending on Maduro's response today, they could impose an oil export embargo that would likely collapse the state.

Instead, U.S. officials hope Maduro heeds the protests, and the military withdraws its support.

Maduro will fight, but he faces a perfect storm of economic pressure, international condemnation, a popular opponent, and the mobilization of his own people.

For the "PBS NewsHour," I'm Nick Schifrin.