The foundation of Euclidean geometry is the concept of a few undefined terms: points, lines, and planes. In essence, a point is an exact position or location on a surface. A point has no actual length or width. A line shows infinite distance and direction but absolutely no width. A line has at least two points lying on it. Euclid’s first postulate is that only one unique straight line can be drawn between any two points. Line segments are lines that have a set length and do not go on forever.
Euclid’s second postulate is that a finite straight line, or line segment, can be extended continuously into a straight line. The last of Euclid’s undefined terms is a plane, a flat surface similar to a table top or floor. However, a plane’s area is infinite. It has never ending length and width but has no depth. Lines can intersect each other or they can be parallel. Intersecting lines can be perpendicular, meaning they cross at a right angle. Lines in a plane that do not intersect or touch at a point and have a constant, unchanging distance between each other are called parallel lines.
Line segments can be used to create different polygons. As in Euclid’s third postulate, with any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. All the angles in a triangle add up to 180 degrees. An acute angle is less than 90 degrees. A right angle is 90 degrees; all right angles are equal, as stated in Euclid’s fourth postulate. An obtuse angle is greater than 90 degrees but less than 180 degrees. Lastly, 180 degrees makes up a straight line. Two triangles with the same angles are not necessarily congruent.
The Term Paper on Is Having a Degree Important Pros and Cons
I got the idea for this article after overhearing a conversation the other day between two teenagers. One was arguing that getting a degree was the only way to get a good job to make good money and the other was arguing that getting a degree wasn’t necessary in today’s day and age in order to make good money. I’m sure you or someone you know have had this debate at least once and I think it’s one ...
Today we know the fifth postulate as the rule that through any point that is not on a line, there is only one line that is parallel to the line. One kind of Non-Euclidean Geometry is Riemannian, or elliptic, geometry. It is a geometry based on curved/spherical, surfaces invented by a German man named Bernhard Riemann. In 1889 he rediscovered the work of an Italian mathematician which stated certain problems in Euclidean Geometry. The earth is round and spherical so elliptic geometry is useful, and directly related to our everyday lives. Elliptical geometry is not limited to spheres and can be used applied on cylinders (Roberts).
Euclid’s first postulate is false in elliptic geometry. Between two points there are many different lines that will connect them. The shortest distance between two points is called a minimal geodesic. Also, because it is based on curved surfaces, straight lines are impossible. This makes Euclid’s second postulate untrue as well. If you extend a line on a sphere or cylinder, most of the time the line will curve back around and form a circle. A line will always curve in elliptic geometry. In elliptic geometry all the angles in a triangle add up to greater than 180 degrees.
Two triangles with the same angles are not just similar, they are actually congruent. In Elliptic there are no parallel lines (Elliptic geometry).
Another kind of Non-Euclidean Geometry is Lobachevsky, or Hyperbolic, Geometry. It is also called Lobachevsky-Bolyai-Gauss (Weisstein).
It is a geometry based on saddle-shaped space, similar to a Pringle. Hyperbolic geometry was invented by a Russian mathematician named Nicholas Lobachevsky. Lobachevsky also expanded on Euclid’s ideas. It’s very hard to see how this geometry is useful but it can be used in gradational fields, space travel, and astronomy (Roberts).
