GEOMETRY

PARALLEL LINES AND TRANSVERSALS

• Parallel Lines: are coplanar lines that have no points in common
• Transversal: is a line that intersects two other coplanar lines
• Interior Angles: angles in-between the parallel lines.
• Exterior Angles: angles outside the parallel lines.
• Alternate Angles: angles on opposite sides of the transversal.

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THEOREMS

• Alternate Interior Angles: pairs of angles that are in-between the parallel
• lines AND on opposite sides of the transversal. MEASURES ARE EQUAL
• Alternate Exterior Angles: pairs of angles that are outside the parallel
• lines AND on opposite sides of the transversal. MEASURES ARE EQUAL

Geometry has a long line in history and first started out with no name to refer to. It already existed in 3000 B.C in ancient civilizations of Babylon and Egypt.

Euclid of Alexandria wrote a book called "elements", which now became our modern geometry. The geometry use today referred as the Euclidean Geometry.

Euclid is considered to be the "Father of Geometry", but modern geometry started in the end of the 16 th century with the arrival of analytic geometry.

Geometry that wasn't Euclidean came into the 19th century as problems with several of Euclid's postulates surfaced.

The geometry taught today in universities and colleges are directly credited to Euclid for his organization in ancient knowledge.

It is only proper to consider Euclid as the inventor of modern geometry. Basic geometry has been around since 3000 B.C, but Geometry was formed 300 BC.

Since geometry has been around in one form or another, it is hard to say where geometry originated.

The Egyptians were using geometry to build pyramids by where to put the bricks and what angle it should be. If anything was out of line by an inch the whole thing would fall down.

Also, the Greeks began to carefully study geometry and tried to prove facts about it. Euclid was perhaps the most notable geometry of ancient mathematics and he was the first to theorize the subject.

Ancient Egyptians demonstrated knowledge of Geometry by surveying and constructing. The Nile River overflowed so it constantly had to be surveyed. Babylonians had tablets that revealed they were conscious of the Pythagorean Theorem.

Ancient Greeks practiced centuries of experimental geometry like Egypt and Babylonia had, and they absorbed the experimental geometry of both of those cultures. Then they created the first formal mathematics of any kind by organizing geometry with rules of logic.

Euclid's (400BC) important geometry book The Elements formed the basis for most of the geometry studied in schools ever since. There are two main types of mathematical (including geometric) rules : postulates (also called axioms), and theorems.

Postulates are basic assumptions - rules that seem to be obvious and are therefore accepted without proof. Theorems are rules that must be proved.

Euclid gave five postulates. The fifth postulate reads: Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.

Euclid was not satisfied with accepting the fifth postulate (also known as the parallel postulate) without proof. Many mathematicians throughout the next centuries unsuccessfully attempted to prove Euclid's Fifth.

It seems to have been known from most ancient of times that the ratio of the circumference and diameter of a circle is a constant, but what is that constant? A search for a better answer to that question has intrigued mathematicians throughout history.

Descartes made one of the greatest advances in geometry by connecting algebra and geometry. A myth is that he was watching a fly on the ceiling when he conceived of locating points on a plane with a pair of numbers.

Maybe this has something to do with the fact that he stayed in bed everyday until 11:00 A.M. Fermat also discovered coordinate geometry, but it's Descartes' version that we use today.

DIFFERENT TYPES OF GEOMETRY

• Non-Euclidean Geometries
• Differential Geometry
• Fractal Geometry

Since mathematicians couldn't prove the 5th postulate, they devised new geometries with "strange" notions of parallelism. (A geometry with no parallel lines?!?) Bolyai and Lobachevsky are credited with devising the first non-euclidean geometries.

Differential geometry combines geometry with the techniques of calculus to provide a method for studying geometry on curved surfaces. Gauss and Riemann (his student) laid the foundation of this field. Einstein credits Gauss with formulating the mathematical fundamentals of the theory of relativity.

Fractals are geometric figures that model many natural structures like ferns or clouds. The invention of computers has greatly aided the study of fractals since many calculations are required. Mandelbrot is one of the researchers of fractal geometry. The word ‘geometry’ comes from the Greek words ‘geo’, meaning earth, and ‘metria’, meaning measure.

Along with arithmetic, geometry was one of the two fields of pre-modern mathematics.

Ancient Egyptians used geometry principles as far back as 3000 BC, using equations to approximate the area of circles among other formulas.

Babylonians measured the circumference of a circle as approximately 3 times the diameter, which is fairly close to today’s measurement which uses the value of Pi (around 3.14).

A Greek mathematician named Euclid who lived around the year 300 BC is often referred to as the ‘Father of Geometry’ for his amazing geometry works that included the influential ‘Elements’, which remained the main textbook for teaching mathematics until around the early 20th century.

Greeks constructed aesthetically pleasing buildings and artworks based on the golden ratio of approximately 1.618.

Greek philosopher and mathematician Pythagoras lived around the year 500 BC and is known for his Pythagorean theorem relating to the three sides of a right angle triangle: a² + b² = c²

Archimedes of Syracuse lived around the year 250 BC and played a large role in the history of geometry including a method for determining the volume of objects with irregular shapes.

The compass and straight edge were powerful tools in the advancement of geometry, allowing the construction of various lengths, angles and geometric shapes.

Modern day geometry has made developments in a number of areas, including those that make use of the raw computing power of today’s computers. Geometry's origins go back to approximately 3,000 BC in ancient Egypt. Ancient Egyptians used an early stage of geometry in several ways, including the surveying of land, construction of pyramids, and astronomy. Around 2,900 BC, ancient Egyptians began using their knowledge to construct pyramids with four triangular faces and a square base.

Euclid's Elements
The next great advancement in geometry came from Euclid in 300 BC when he wrote a text titled 'Elements.' In this text, Euclid presented an ideal axiomatic form (now known as Euclidean geometry) in which propositions could be proven through a small set of statements that are accepted as true. In fact, Euclid was able to derive a great portion of planar geometry from just the first five postulates in 'Elements.' These postulates are listed below:

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• A straight line segment can be drawn joining any two points.
• A straight line segment can be drawn joining any two points.
• All right angles are congruent.

René Descartes' Coordinate Geometry
The next tremendous advancement in the field of geometry occurred in the 17th century when René Descartes discovered coordinate geometry. Coordinates and equations could be used in this type of geometry in order to illustrate proofs. The creation of coordinate geometry opened the doors to the development of calculus and physics.

The Development of Non-Euclidean Geometry
In the 19th century, Carl Friedrich Gauss, Nikolai Lobachevsky, and János Bolyai formally discovered non-Euclidean geometry. In this kind of geometry, four of Euclid's first five postulates remained consistent, but the idea that parallel lines do not meet did not stay true. This idea is a driving force behind elliptical geometry and hyperbolic geometry.

THE END