Spherical Geometry

Spherical geometry is indeed an example of non-Euclidean geometry. Unlike Euclidean geometry, which deals with flat surfaces and straight lines, spherical geometry is concerned with objects on the surface of a sphere. In spherical geometry, the sum of angles in a triangle is always greater than 180 degrees, and there are no parallel lines. This deviation from the principles of Euclidean geometry makes spherical geometry a non-Euclidean system. Therefore, the statement "Spherical Geometry is an example of non-Euclidean geometry" is true.

Infinitely many great circles can be drawn through the poles on a sphere because the poles are the points where the axis of rotation intersects the surface of the sphere. Any plane passing through the axis of rotation will intersect the sphere in a great circle, and since there are infinitely many planes that can be defined by the axis of rotation, there are also infinitely many great circles that can be drawn through the poles.

Vertical angles are formed when two lines intersect. In both Euclidean and spherical geometry, when two lines intersect, the angles that are opposite each other are called vertical angles. These vertical angles are congruent, meaning they have the same measure. Therefore, the statement that "vertical angles are congruent in both Euclidean and spherical geometry" is true.

A right angle is defined as an angle that measures exactly 90 degrees. In a triangle, the sum of the three angles is always 180 degrees. If a triangle has more than one right angle, it means that the sum of the angles would be greater than 180 degrees, which is not possible. Therefore, a triangle cannot have more than one right angle. Hence, the given answer "True" is incorrect.

Parallel lines cannot be drawn on a sphere. In Euclidean geometry, parallel lines are lines that never intersect. However, on a sphere, any two lines will eventually intersect at two points, which means they are not parallel. Therefore, the statement is false.

A great circle on a sphere is a circle that has the same center as the sphere and divides it into two equal halves. The dotted line shown on the sphere in the question does not meet this criteria, as it does not divide the sphere into two equal halves. Therefore, the statement that the dotted line is a great circle is false.

On a sphere, two lines can intersect at zero points (if they are parallel and do not intersect the sphere) or at two points (if they are not parallel and intersect the sphere at two different points). Therefore, the statement that two lines intersect at one and only one point on a sphere is false.

The sum of the angles of a triangle on a sphere can be at most 540 degrees. This is because the sum of the angles in a triangle on a flat plane is always 180 degrees. However, on a sphere, the angles of a triangle can be greater than 180 degrees. In fact, the sum of the angles in a triangle on a sphere is directly proportional to its surface area. Since the maximum surface area of a triangle on a sphere is 4π (a hemisphere), the sum of the angles can be at most 540 degrees.

When two lines intersect on a sphere, eight angles are made. This is because when two lines intersect, they form four pairs of opposite angles. On a sphere, each pair of opposite angles is duplicated on the other side of the sphere, resulting in a total of eight angles.

The shortest distance between two points on a sphere is along the arc of a great circle. A great circle is a circle on a sphere that has the same center as the sphere. The arc of a great circle is the portion of the circle that connects two points on the sphere. This path is the shortest distance because it follows the curvature of the sphere, taking into account the spherical geometry. A triangle on a sphere would not represent the shortest distance between two points.

In spherical geometry, two lines that are perpendicular to the same line are not parallel to each other. In fact, in spherical geometry, there are no parallel lines. This is because in spherical geometry, lines are represented as great circles, which are circles that have the same center as the sphere. Any two great circles on a sphere will intersect at two points, meaning they are not parallel. Therefore, the statement "Two lines that are perpendicular to the same line are parallel to each other" is not true in spherical geometry.

Perpendicular lines on a sphere form 8 right angles because each line that is perpendicular to the surface of the sphere intersects with the surface at two points, creating two right angles at each point of intersection. Since there are four points of intersection for each line, there are a total of 8 right angles formed by perpendicular lines on a sphere.

A great circle is a circle formed by the intersection of a sphere and a plane that passes through the center of the sphere. Since all lines of longitude pass through the poles and the center of the Earth, they intersect the Earth's surface along great circles. Therefore, the statement that all lines of longitude are great circles is true.

In spherical geometry, a line does not have infinite length. Instead, a line in spherical geometry is defined as a great circle, which is a circle on the sphere that has the same center as the sphere. A great circle has a finite length, just like any other circle. Therefore, the statement that a line in spherical geometry has infinite length is incorrect.