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Spherical triangles are a fascinating concept in geometry, especially when you're dealing with a three-dimensional surface like a globe. Unlike the flat-plane triangles you're used to in traditional geometry, spherical triangles are formed on the surface of a sphere. Imagine a triangle drawn on the surface of the Earth, where each side arches along the globe's surface. These sides are not straight lines but arcs that are part of what we call great circles.

A spherical triangle has three vertices and three sides, just like a regular triangle, but its sides are composed of arcs that make it unique. It's essential to understand this difference because the rules that apply to flat triangles change when they're on a sphere. One significant difference is that the sum of the angles in a spherical triangle is more than 180 degrees, which can be quite surprising if you're only familiar with flat geometry.

In the exercise, the vertices of the spherical triangle are Wellington, Moscow, and Rio de Janeiro, creating a broad triangle whose sides hug the Earth’s surface closely. This underscores the key concept of spherical triangles: they give us an entirely new way of understanding and measuring the geometry on spherical surfaces.

In spherical geometry, the sum of the angles of a triangle is one of the most intriguing properties. While we're used to the idea that the angles of a triangle add up to precisely 180 degrees, spherical triangles break this rule due to the curved nature of their surface.

When you're on a sphere, the sum of the angles is always greater than 180 degrees. This is because each angle is formed at the intersection of arcs of great circles, which are larger than the angles in a flat space. The increase in the sum of these angles over 180 degrees is called spherical excess. This excess gives insight into the triangle's size and shape on the sphere.

This excess can vary, depending on how large the triangle is on the sphere, even reaching up to 540 degrees. So, when you form a triangle between Wellington, Moscow, and Rio de Janeiro, you will measure angles that, together, amount to more than 180 degrees, showcasing how brilliant and surprising spherical geometry really is.

Great circles play a crucial role in spherical geometry. They are the largest possible circles that can be drawn on the surface of a sphere and represent the shortest path between two points. In essence, any triangle on a sphere, like the one between Wellington, Moscow, and Rio de Janeiro, is made up of arcs of these great circles.

Each great circle divides the sphere into two equal halves, much like how a flat line can split a flat surface into two. To visualize this, imagine the Earth's equator, which is a classic example of a great circle. All longitudes are also great circles as they run from the North Pole to the South Pole across the globe.

The concept of great circles is valuable in navigation and geography because it helps calculate the most efficient paths over long distances. In the exercise, when you connect Wellington, Moscow, and Rio de Janeiro with great circle arcs, you are essentially forming a path that planes might take when flying between these cities, highlighting the practicality of spherical geometry.