Which of the following statements from Euclidean geometry is also true of spherical geometry?


A) A line has infinite length.
B) Two intersecting lines divide the plane into four regions.
C) Two perpendicular lines create four right angles.
D)The intersection of two lines creates four angles.
Im realy having trouble on this quesiton & I need some help:) thankyou!

Abyo

Sahrra

To determine which statement from Euclidean geometry is also true of spherical geometry, let's go through each option and analyze them one by one:

A) A line has infinite length: This statement is true in both Euclidean and spherical geometries. In Euclidean geometry, a line extends infinitely in both directions. Similarly, in spherical geometry, a line (great circle) on the surface of a sphere can be seen as extending infinitely in both directions.

B) Two intersecting lines divide the plane into four regions: This statement is NOT true in spherical geometry. In Euclidean geometry, two intersecting lines form four regions, but in spherical geometry, two intersecting great circles divide the sphere's surface into two regions instead. These regions are separated by the two great circles.

C) Two perpendicular lines create four right angles: This statement is NOT true in spherical geometry. In Euclidean geometry, two perpendicular lines intersect at a right angle, forming four right angles. However, in spherical geometry, two perpendicular great circles intersect at two points, resulting in only two right angles instead of four.

D) The intersection of two lines creates four angles: This statement is true in both Euclidean and spherical geometries. When two lines intersect in Euclidean geometry, they create four angles around the intersection point. Similarly, in spherical geometry, when two great circles intersect on the surface of a sphere, they also create four angles around the intersection points.

Therefore, the correct answer is option D) "The intersection of two lines creates four angles." This statement is true in both Euclidean and spherical geometries.

Think of great circles on a sphere. Those correspond to straight lines in a plane. An arc of a great circle is the shortest distance between two points, measured along the surface of the sphere. A great circle is one whose diameter is the same as the diameter of the sphere.

not A: On a sphere, a great circle does not have infinite length.

not B: Anything pertaining to a plane has no relevance on a sphere.

not C: Two perpendicular great circles intersect twice. Think of the equator and a meridian. At each intersection there are 4 right angles.

D: This is true. Although as in C, two "lines" on a sphere always intersect twice.

By the way, that means that in spherical geometry, there are no parallel lines!