EUCLID

1. 1
EUCLIDEAN
GEOMETRY
EUCLIDEAN
GEOMETRY
The Greeks organized the
Mathematical Properties into an
Axiomatic System, now known
as Euclidean Geometry.
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What is Euclidean Geometry?
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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.
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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.
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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:
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Second Postulate
2. To produce a finite straight line
continuously in a straight line.
That is, we can extend the line
indefinitely.

2. 2
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Third Postulate
3. To describe a circle with any center and
distance.
That is, circle exists.
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Fourth Postulate
4. That all right angles are equal to one
another.
90◦
90◦
90◦
90◦
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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.
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Essential Question
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Visualisation
Point
Line
Ray
Plane
Endpoints
Line Segment
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Visualisation
Point
Line
Ray
Plane
Endpoints
Line Segment
POINT
LINE
SEGEMENT
RAY

3. 3
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Point
A dimensionless geometric object having
no properties except location
an entity that has a location in space but
no extent
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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.
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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.
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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.
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Essential Question
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Parallel Lines
Two lines in a plane that do not
intersect are called parallel lines.

4. 4
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Parallel lines
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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.
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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.
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Angles
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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.
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Measurement of Angle
Right Angle
ACUTE ANGLE
OBTUSE ANGLE

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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.
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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
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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
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Reflection
60 degrees
60 degrees
reflected light ray'Incomming light ray
Reflective Surface
Normal Line
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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.
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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.

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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
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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◦
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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.
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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.
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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:
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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.

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Planes Intersecting
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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
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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
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References
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