1. Euclid's Elements/Postulates - Euclid wrote a text titled 'Elements' in 300 BC which presented geometry through a small set of statements called postulates that are accepted as true. He was able to derive much of planar geometry from just five postulates, including the parallel postulate which caused much debate. 2. Euclid's Contribution to Geometry - Euclid is considered the "Father of Geometry" for his work Elements, which introduced deductive reasoning to mathematics. Elements influenced the development of the subject through its logical presentation of geometry from definitions and postulates. 3. Similar Triangles - Triangles are similar if they have the same shape but not necessarily the same

1. LovelyAnnD.R.Caluag
BS Math (BA) 3-B
BulacanState University
Egyptians c. 2000 - 500 B.C.
Ancient Egyptians demonstrated a practical knowledge of geometry through
surveying and construction projects. The Nile River overflowed its banks
every year, and the river banks would have to be re-surveyed. See a PBS
Nova unit on those big pointy buildings. In the Rhind Papyrus, pi is
approximated.
Babylonians c. 2000 - 500 B.C.
Ancient clay tablets reveal that the Babylonians knew the Pythagorean
relationships. One clay tablet reads
4 is the length and 5 the diagonal. What is the breadth? Its size is not
known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there
remains 9. What times what shall I take in order to get 9? 3 times 3 is 9. 3
is the breadth.
Greeks c. 750-250 B.C.
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.
The Fifth Postulate
Controversy
c. 400 B.C. - 1800 A. D.
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.
The Search for pi ??? B.C. - present
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.
Coordinate
Geometry
c. 1600 A.D.
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.

2. Non-Euclidean
Geometries
c. early 1800's
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
c. late 1800's-1900's
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.
Fractal Geometry c. late 1800's-1900's
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.
Source: http://math.rice.edu/~lanius/Geom/his.html
2. Euclid's Elements/Postulates
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:
(1) A straight line segment can be drawn joining any two points.
(2) A straight line segment can be drawn joining any two points.
(3) Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
(4) All right angles are congruent (the same).
(5) If two lines are drawn which intersect a third line in such a way that the sum of the inner angles on one side is less than
two right angles, then the two lines inevitably must intersect each other on that side if extended infinitely.
Euclid's fifth postulate is also known as the parallel postulate.
Source: http://www.wyzant.com/resources/lessons/math/geometry/introduction/history_of_geometry
3. Euclids Contribution to Geometry
Euclid
Euclid (yōˈklĭd) [key], fl. 300 B.C., Greek mathematician. Little is known of his life other than the fact that he taught at
Alexandria, being associated with the school that grew up there in the late 4th cent. B.C. He is famous for his Elements, a
presentation in thirteen books of the geometry and other mathematics known in his day. The first six books cover elementary
plane geometry and have served since as the basis for most beginning courses on this subject. The other books of
the Elements treat the theory of numbers and certain problems in arithmetic (on a geometric basis) and solid geometry,
including the five regular polyhedra, or Platonic solids. A few modern historians have questioned Euclid's authorship of
the Elements, but he is definitely known to have written other works, most notably the Optics.
The great contribution of Euclid was his use of a deductive system for the presentation of mathematics. Primary terms, such
as point and line, are defined; unproved assumptions, or postulates, regarding these terms are stated; and a series of
statements are then deduced logically from the definitions and postulates. Although Euclid's system no longer satisfies
modern requirements of logical rigor, its importance in influencing the direction and method of the development of
mathematics is undisputed.
One consequence of the critical examination of Euclid's system was the discovery in the early 19th cent. that his fifth
postulate, equivalent to the statement that one and only one line parallel to a given line can be drawn through a point external
to the line, can not be proved from the other postulates; on the contrary, by substituting a different postulate for this parallel
postulate two different self-consistent forms of non-Euclidean geometry were deduced, one by Nikolai I. Lobachevsky (1826)
and independently by János Bolyai (1832) and another by Bernhard Riemann (1854).
See D. Berlinski, The King of Infinite Space: Euclid and His Elements (2013).

3. Read more: Euclid, Greek mathematician | Infoplease.com http://www.infoplease.com/encyclopedia/people/euclid-greek-
mathematician.html#ixzz3IjGPq400
Euclid , another name Euclid of Alexandria and also known as Father of Geometry. A Greek mathematician known for the
major contribution on geometry. The whole new stream of geometry established by him known as Euclidean Geometry.
Basically the modern 2 dimension geometry (mean what u can draw on a paper) is actually adopted from the Euclidean
Geometry. and also it is the basic building block of the modern geometry.
All of his invents and developments , axioms etc. are written on his book "Euclid's Elements" that consisting 13 book. This is
one of the most influential and successful textbook ever written.
He also works of Number Theory, Spherical Geometry, Conic Section etc.
4. Lines and all the subsets of Lines
A line is a collection of points along a straight path that goes on and on in opposite directions. A line has no endpoints.
A line segment is a part of a line having two endpoints.
A line segment is a piece of a line that has two distinct endpoints. Because of these endpoints, unlike a line, a line segment
doesn't extend infinitely. Rather, it is finite, with a measurable length.
Read more : http://www.ehow.com/info_8212934_subsets-line-geometry.html
A ray is a part of a line with one endpoint and goes on and on in one direction...
A ray is essentially a hybrid between a line and a line segment. A ray has exactly one endpoint -- called its origin -- and extends
infinitely in the other direction. Like lines, rays are infinite and therefore immeasurable. Rays may sometimes be referred to
as half-lines.
5. Triangles
A triangle has three sides and three angles.
The three angles always add to 180°
Equilateral, Isosceles and Scalene
There are three special names given to triangles that tell how many sides (or angles) are equal.
There can be 3, 2 or no equal sides/angles:
Equilateral Triangle
Three equal sides
Three equal angles, always 60°
Equilateral Triangle
Three equal sides
Three equal angles, always 60°
Scalene Triangle
No equal sides
No equal angles
What Type of Angle?
Triangles can also have names that tell you what type of angle is inside:
Acute Triangle
All angles are less than 90°
Right Triangle
Has a right angle (90°)
Obtuse Triangle
Has an angle more than 90°
Combining the Names
Right Isosceles Triangle
Has a right angle (90°), and also two equal angles
6. Similar Triangles
Definition: Triangles are similar if they have the same shape, but can be different sizes.
(They are still similar even if one is rotated, or one is a mirror image of the other).
Triangles are similar if they have the same shape, but not necessarily the same size. You can think of it as "zooming in" or out
making the triangle bigger or smaller, but keeping its basic shape.
Properties of Similar Triangles
1. Corresponding angles are congruent (same measure)
2. Corresponding sides are all in the same proportion
How to tell if triangles are similar

4. Any triangle is defined by six measures (three sides, three angles). But you don't need to know all of them to show that two
triangles are similar. Various groups of three will do. Triangles are similar if:
AAA (angle angle angle)
All three pairs of corresponding angles are the same.
SSS in same proportion (side side side)
All three pairs of corresponding sides are in the same proportion
SAS (side angle side)
Two pairs of sides in the same proportion and the included angle equal.