Euclidean geometry is based on Euclid's postulates and describes geometry in two or three dimensions. It includes concepts like points, lines, planes, angles and their properties. Some key principles are: parallel lines never intersect; the internal angles of a triangle sum to 180 degrees; and the Pythagorean theorem relating the sides of a right triangle. Euclidean geometry provides an axiomatic framework to define and prove theorems about geometric shapes and their relationships using logic and definitions. It has applications in fields like engineering and remains important in mathematics.
Inscribe Circles in Triangles using Geometric ConstructionColin
Inscribe circles in triangles using geometric construction, with PC geometry software such as Dr Geo on Linux. This slide show presents some innovative constructions. The slide show and included instructions are public domain. Basic use of compass and straightedge is advised as a prerequisite topic.
Inscribe Circles in Triangles using Geometric ConstructionColin
Inscribe circles in triangles using geometric construction, with PC geometry software such as Dr Geo on Linux. This slide show presents some innovative constructions. The slide show and included instructions are public domain. Basic use of compass and straightedge is advised as a prerequisite topic.
Circles in Triangles using Geometric ConstructionColin
Inscribe circles in triangles using geometric construction, with PC geometry software such as Dr Geo on Linux. This slide show presents some innovative constructions. The slide show and included instructions are public domain. Basic use of compass and straightedge is advised as a prerequisite topic.
A Presentation on the Geometric Bonanza. Hope it will be helpful to students in search of this Topic and even in the topic of Geometry and its Applications. Hope u enjoy it.
SHARIGUIN PROBLEMS IN PLANE GEOMETRY
This volume contains over 600 problems in plane geometry and consists of two parts. The first part contains rather simple problems to be solved in classes and at home. The second part also contains hints and detailed solutions. Over 200 new problems have been added to the 1982 edition, the simpler problems in the first addition having been eliminated, and a number of new sections- (circles and tangents, polygons, combinations of figures, etc.) having been introduced, The general structure of the book has been changed somewhat to
accord with the new, more detailed, classification of the problems. As a result, all the problems in this volume have been rearranged.
Circles in Triangles using Geometric ConstructionColin
Inscribe circles in triangles using geometric construction, with PC geometry software such as Dr Geo on Linux. This slide show presents some innovative constructions. The slide show and included instructions are public domain. Basic use of compass and straightedge is advised as a prerequisite topic.
A Presentation on the Geometric Bonanza. Hope it will be helpful to students in search of this Topic and even in the topic of Geometry and its Applications. Hope u enjoy it.
SHARIGUIN PROBLEMS IN PLANE GEOMETRY
This volume contains over 600 problems in plane geometry and consists of two parts. The first part contains rather simple problems to be solved in classes and at home. The second part also contains hints and detailed solutions. Over 200 new problems have been added to the 1982 edition, the simpler problems in the first addition having been eliminated, and a number of new sections- (circles and tangents, polygons, combinations of figures, etc.) having been introduced, The general structure of the book has been changed somewhat to
accord with the new, more detailed, classification of the problems. As a result, all the problems in this volume have been rearranged.
Chapter 1 ( Basic Concepts in Geometry )rey castro
Chapter 1 Basic Concepts in Geometry
1.1 Points, Lines and Planes
1.2 Line Segment
1.3 Rays and Angles
1.4 Some Special Angles
1.5 Angles Made By A Transversal
1.6 Transversal Across Two Parallel Lines
1.7 Conditions For Parallelism
It helps to know about angles.It will also help them in their studies.To know the interesting and they will be inter acted to the studies.It will get idea how it is prepared and they will also try to make it
This powerpoint includes:
Triangles and Quadrangles
Definition, Types, Properties, Secondary part, Congruency and Area
Definitions of Triangles and Quadrangles
Desarguesian Plane
Mathematician Desargues and His Background
Harmonic Sequence of Points/Lines
Illustrations and Animated Lines.
Eucluidian and Non eucluidian space in Tensor analysis.Non Euclidian space AJAY CHETRI
Eucluidian and Non eucluidian space in Tensor analysis.
Introduction to type of system in sphere.Benefit and advantage of using Tensor analysis.EUCLID’S GEOMETRY
VS.
NON-EUCLIDEAN GEOMETRY
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
012 euclidean geometry[1]
1. 1
EUCLIDEAN
GEOMETRY
EUCLIDEAN
GEOMETRY
The Greeks organized the
Mathematical Properties into an
Axiomatic System, now known
as Euclidean Geometry.
2
What is Euclidean Geometry?
3
What is Euclidean Geometry?
The geometry (plane and solid) based on
Euclid's postulates.
In mathematics, Euclidean geometry is
the familiar kind of geometry on the
plane or in three dimensions.
Mathematicians sometimes use the term
to encompass higher dimensional
geometries with similar properties.
4
Euclid Five Postulates
1. Any two points can be joined by a straight line.
2. Any straight line segment can be extended
indefinitely in a straight line.
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.
5. Parallel postulate. If two lines intersect a third 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 far enough.
5
First Postulate
1. To draw a straight line from any point to
any point.
That is, we can draw one unique
straight line through two distinct points:
6
Second Postulate
2. To produce a finite straight line
continuously in a straight line.
That is, we can extend the line
indefinitely.
2. 2
7
Third Postulate
3. To describe a circle with any center and
distance.
That is, circle exists.
8
Fourth Postulate
4. That all right angles are equal to one
another.
90◦
90◦
90◦
90◦
9
Fifth or Parallel Postulate
α
β
α+β < 180◦
α
β
α+β = 180◦
Never intersection
The statement of
the fifth postulate is
complicated. Many
attempted to prove
the 5th from the first
four but failed.
10
Essential Question
11
Visualisation
Point
Line
Ray
Plane
Endpoints
Line Segment
12
Visualisation
Point
Line
Ray
Plane
Endpoints
Line Segment
POINT
LINE
SEGEMENT
RAY
3. 3
13
Point
A dimensionless geometric object having
no properties except location
an entity that has a location in space but
no extent
14
Line
A geometric figure formed by a point moving
along a fixed direction and the reverse
direction.
A line can be described as an infinitely thin,
infinitely long, perfectly straight curve (the term
curve in mathematics includes "straight
curves"). In Euclidean geometry, exactly one
line can be found that passes through any two
points. The line provides the shortest
connection between the points.
15
Line Segment & Endpoints
Line Segment are formed by joining
two points (in the shortest possible
way or is the part of a line lying
between two points on that line.
These two points are called
endpoints.
16
Ray
A Ray is the part of a line lying on
one side of a point on the line.
A ray starts at one point, then goes
on forever in one direction.
17
Essential Question
18
Parallel Lines
Two lines in a plane that do not
intersect are called parallel lines.
4. 4
19
Parallel lines
20
Angles
An angle is the union of two line
segments with a common endpoint
called a vertex.
Used to represent an amount of
rotation (turning) about a fixed point
in counterclockwise direction.
21
Angles
Suppose there are two rays with a
common endpoint. The two rays and the
region between them is called the angle
at a point P formed by the two rays.
The smallest amount of
counterclockwise rotation about P
needed to rotate one of the rays to the
position of the other ray.
22
Angles
23
Measurement of Angle
Degrees – indicated with a little
circle: º. For example 90º. A full
circle (to come back where you
started) is 360º. Half turn is 180º.
Clockwise turns have a negative
measurement.
24
Measurement of Angle
Right Angle
ACUTE ANGLE
OBTUSE ANGLE
5. 5
25
Perpendicular Lines
Right Angle – If the angle formed by
the two rays is 90º.
When two lines in a plane meet,
they form four angles.
When all four of these angles made
by two intersecting lines are 90º, the
lines are called perpendicular lines.
26
Normal Line
Normal Line at a point on surface is a
line that passes through that point and is
perpendicular to the surface at that point.
Physical Principle of reflection –
Incoming light and reflected light make
the same angle with the normal line at
the point where the incoming light ray
hits the surface
27
Reflection
Normal Light ray lies in the
same plane as the normal line
and the incoming light ray.
The reflected ray and incoming
light ray coincide only when
incoming light ray lines up with
the normal line
28
Reflection
60 degrees
60 degrees
reflected light ray'Incomming light ray
Reflective Surface
Normal Line
29
Common Notions (Axioms)
1. Things which are equal to the same thing are
also equal to one another. [a=c, b=c => a =
b]
2. If equals be added to equals, the wholes are
equal. [a=b => a+c = b+c]
3. If equals be subtracted from equals, the
remainders are equal. [a=b => a-c = b-c]
4. Things which coincide with one another are
equal to one another.
5. The whole is greater than the part.
30
Thales’ theorem of “vertical
angles are equal”
a b
c
Straight line spans an angle
of 180◦, so
a + c = 180◦, c + b = 180◦
By common notation 1, we
have
a + c = c + b
By common notion 3, we
subtract c from above,
getting
a = b.
Pair of non-adjacent
angles a and b are
called vertical angles,
prove a = b.
6. 6
31
Theorem of Transversal Angles
b
c
a
The transverse line with two
parallel lines makes angles a and
b. Show a = b = c.
From the vertical
angle theorem, c = b.
Clearly, c + d = 180◦,
a + d = 180◦ (parallel
postulate), so
a = c = b.
d
32
Angle Sum Theorem
a b
c
Show the angle sum of a
triangle is
a + b + c = 180◦
ba
Draw a line through
the upper vertex
parallel to the base.
Two pairs of
alternate interior
angles are equal,
from previous
theorem. It follows
that
a + b + c = 180◦
33
Side-Angle-Side Theorem
α
a
b
α
a
b
If two triangles have equal
lengths for the corresponding
side, and equal angle for the
included angle, then two
triangles are “congruent”.
That is, the two triangles can
be moved so that they overlap
each other.
34
Pythagoras Theorem
a
b
c
Show the sides of a
right triangle satisfies
a2 + b2 = c2
On the sides of a square, draw
alternatively length a and b. Clearly,
all the triangles are congruent by the
side-angle-side theorem. So the four
lengths inside the outer square are
equal. Since the sum of three angles
in a triangle is 180◦, we find that the
inner quadrilateral is indeed a square.
Consider two ways of computing the
area:
(a+b)2 = a2 + 2ab + b2,
And c2 + 4 ( ½ ab) = c2+2ab.
They are equal, so a2+b2 = c2.
35
Line Equation
Lines in a Cartesian plane can be
described algebraically by linear
equations and linear functions. In two
dimensions, the characteristic equation
is often given by the slope-intercept
form:
36
Plane
A surface containing all the straight lines that
connect any two points on it.
A plane is a surface such that, given any three
points on the surface, the surface also
contains the straight line that passes through
any two of them. One can introduce a
Cartesian coordinate system on a given plane
in order to label every point on it uniquely with
two numbers, the point's coordinates.
7. 7
37
Planes Intersecting
38
Within any Euclidean space, a plane is
uniquely determined by any of the
following combinations:
three non-collinear points (not lying on
the same line)
a line and a point not on the line
two different lines which intersect
two different lines which are parallel
Properties of Euclidean Space
39
In three-dimensional Euclidean space, we may
exploit the following facts that do not hold in
higher dimensions:
Two planes are either parallel or they intersect
in a line.
A line is either parallel to a plane or they
intersect at a single point.
Two lines normal (perpendicular) to the same
plane must be parallel to each other.
Two planes normal to the same line must be
parallel to each other.
Properties of Euclidean Space
40
References
http://www.daviddarling.info/childrens_en
cyclopedia/Moon_Chapter2.html