Suppose you measure the angles of a triangle and find that they add to 185 degrees. From this you can determine that the space the triangle occupies is a. flat. b. positively curved. c. negatively curved. d. filled with dark matter.

In geometry, a positively curved space, like the surface of a sphere, is one where the shortest distance between two points is an arc of a circle rather than a straight line. The key idea here is that the space curves back on itself. You can imagine blowing up a balloon—its surface is a positively curved space. Unlike flat or Euclidean space, which is straightforward and linear, positively curved spaces introduce unique properties for lines and shapes.
One important feature of positively curved spaces is that parallel lines eventually converge. This is unlike Euclidean geometry, where parallel lines never meet. This type of curvature is instrumental in understanding phenomena in both small scales (like on a globe) and even cosmological scales, like the curvature of the universe.
This curvature affects the angles within shapes, such as triangles, explained in the sections below.

In Euclidean geometry, the sum of the internal angles of a triangle is always 180 degrees. This concept changes in non-Euclidean geometries like spherical geometry. When dealing with positively curved space, the sum of the angles of a triangle is always greater than 180 degrees.
For example, if you draw a triangle on the surface of a sphere (imagine a triangle with one vertex at the North Pole and the other two along the Equator), you'll find that each angle can be 90 degrees, summing to 270 degrees. This is far greater than the 180 degrees you'd expect in flat, Euclidean geometry. The key takeaway is:

• Flat/Euclidean geometry: The sum of angles is exactly 180 degrees.
• Positively curved geometry: The sum of angles is greater than 180 degrees.
• Negatively curved geometry: The sum of angles is less than 180 degrees.

Knowing this helps in identifying the type of space a shape occupies by measuring the angles.

Euclidean geometry, named after the ancient Greek mathematician Euclid, is the geometry of flat spaces. It includes familiar concepts taught in school, like straight lines, circles, squares, and angles.
Some core properties include:

• The sum of the angles in a triangle is always 180 degrees.
• Parallel lines never intersect and are always equidistant.
• The shortest distance between two points is a straight line.

These principles apply to the 'flat' world we experience daily—think of a piece of paper or a flat screen. However, in Euclidean geometry, these properties are constant and do not change regardless of the triangle or parallel lines drawn. This differs significantly when you move to non-Euclidean geometries, where these properties can vary widely due to curvature of space.

Spherical geometry is a type of non-Euclidean geometry specifically concerned with the properties and relationships of points, lines, and shapes on the surface of a sphere. This yields some fascinating differences from Euclidean geometry.
Key concepts include:

• Triangles on a sphere have angle sums greater than 180 degrees.
• Great circles (like the Equator or longitudinal lines on a globe) are the 'straight lines' of spherical geometry.
• Parallel lines do not exist in spherical geometry as all lines (great circles) eventually intersect.

An everyday example of spherical geometry at work is the way airline routes are plotted on a globe. Routes are often curved on a flat map because they follow great circles, the shortest path between two points on a sphere. Understanding spherical geometry is crucial for fields like navigation, astronomy, and even understanding the universe's shape.