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TOOLS OF GEOMETRY
Shirlee Remoto Ocampo
De La Salle University-Manila
Shirlee Remoto Ocampo
“There is no royal road to Geometry.” - Euclid
Let us take a journey…
G E O M E T R Y
Shirlee Remoto Ocampo
Eye and
brain….vision
and mind
working
together?
Shirlee Remoto Ocampo
Are the lines parallel?
Shirlee Remoto Ocampo
Geometric Illusions
how our mind attempts to find orderly representations out of sometimes ambiguous and disorderly 2 dimensional images.
images transmitted from our retina to our brain are imperfect representation of reality
Our visual system is capable of performing complex processing of information received from the eyes in order to extract meaningful perceptions. Sometimes, however, this process can lead to faulty perceptions.
Shirlee Remoto Ocampo
What figure can you see?
Shirlee Remoto Ocampo
Have you tried painting using
points?
Pointillism – art technique of applying dots or colors to a surface so that from there is blending from a distance
Sunday Afternoon on the Island of La Grande Jatte – ten-foot wide painting entirely made of points by Georges Seurat in 1884
Shirlee Remoto Ocampo
TOOLS in Geometry
Points
Lines
Planes
A
B
C l
A B
C
P
Undefined Terms
A definition uses known words to describe a new word.
In geometry, some words such as point, line, and plane are undefined terms. In other words, there is no formal definition for these words, but instead they are explained by using examples and descriptions which allows us to define other geometric terms and properties.
Point
• A point is simply a
location. It has no
dimension (shape or
size), is usually
represented by a small
dot, and named by a
capital letter.
A
Point A
Line
• A line is a set of points
and extends in one
dimension. It has no
thickness or width, is
usually represented by a
straight line with two
arrowheads to indicate
that it extends without
end in both directions,
and is named by two
points on the line or a
lowercase script letter.
A
B
l
Line AB or line l
Plane • A plane is a flat
surface made up of points. It extends in two dimensions, is usually represented by a shape that looks like a tabletop or wall, and is named by a capital script letter or 3 non-collinear points. You must imagine that the plane extends without end, even though the drawing of a plane appears to have edges.
A
B
C
Plane ABC or plane M
Space
• Space is a boundless, three
dimensional set of all points. It
can contain points, lines, and
planes.
Shirlee Remoto Ocampo
Undefined Terms???
Point – no length, no width, no thickness
Line – infinite length, no width, no thickness
Plane – infinite length, infinite width, no thickness
Space – infinite length, infinite width, infinite thickness
Not formally defined in Geometry
Why are they called
‘undefined terms’?
Shirlee Remoto Ocampo
These four things are
called undefined terms because in
geometry these are words that do
not require a formal definition. They
form the building blocks for formally
defining or proving other words and
theorems. These words themselves
are considered so basic that they are
considered to be true without having
to be proved or formally defined.
16
Collinear Points: A collinear set of points is a
set of points all of which lie
on the same straight line.
A B C D
E
•Points A, B, C and D are collinear.
•Points A, E and C are not collinear.
Coplanar points
Coplanar points are points that
lie on the same plane.
Example: Name four points that
are coplanar.
G
D E
H
Shirlee Remoto Ocampo
Solution:
D, E, F, and G lie on the same plane, so they are coplanar. Also D, E, F, and H are coplanar; although, the plane containing them is not drawn.
Line Segment:
Naming a Line Segment:
Use the names of the endpoints. A B
“Line segment” AB is part of “Line” AB
A line segment is the set of
two points on a line called
endpoints, and all points on
the line between the
endpoints.
A B
Congruent Line Segments:
Congruent line segments are
segments that are equal in
measure.
Then AB = CD
Ex: If AB CD , 9 7AB x and 4 13CD x , what is CD? AB?
Ans. CD = 29, AB = 29
Ex
: A
C D
B If AB CD
Midpoint The midpoint of a line segment divides
the line segment into two congruent
segments.
A B M
If M is the midpoint of AB , then:
1)
3)
5)
2)
4)
6)
AM MB AM MB
1
2AM AB
1
2MB AB
2AB AM 2AB MB
Ray
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A ray with endpoint A and containing a second point B, denoted by , is a subset of a line consisting of one endpoint A and all points of the line on one side of the endpoint A.
Lines
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Intersecting lines are coplanar
lines with a common point.
Parallel lines are coplanar lines
that do not intersect.
Concurrent lines are three or
more lines that have a common
point.
Skew lines are lines that do not
lie in the same plane.
POSTULATES on LINES
Postulate 1.1. The Line Postulate
Two points determine a line.
Postulate 1.2. Three noncollinear
points determine a plane.
Postulate 1.3. If two distinct lines
intersect, then they intersect at a
point.
Postulate 1.4. If two distinct points
of a line are in a plane, then the line
is in the plane Shirlee Remoto Ocampo
Postulates on Lines
Postulate 1.5. If two distinct planes
intersect, then they intersect in a
line.
Postulate 1.6. Ruler Postulate
The points of a line can be put
into one-to-one correspondence with
the real numbers so that the distance
between two points is the absolute
value of the difference of the
corresponding numbers. Shirlee Remoto Ocampo
Postulates on Lines
Postulate 1.7. Segment
Addition Postulate
If three points A, B, and C are
collinear, and B is between A and
C, then AB + BC = AC.
Shirlee Remoto Ocampo
Angles and Angle Pairs
An angle is formed by two noncollinear
rays with a common endpoint.
Classifications of Angles:
A right angle is an angle whose measure
is 90o.
An acute angle is an angle whose
measure is less than 90o.
An obtuse angle is an angle whose
measure is greater than 90o and less than
180o.
Shirlee Remoto Ocampo
Postulates on Angles
Postulate 1.8. Angle
Measurement Postulate
To every angle, there corresponds a
real number between 0 and 180.
Postulate 1.9. Angle Construction
PostulateLet be a ray on the edge
of the half-plane H. For every
number r between 0 and 180 there is
exactly one ray , with P in H, such
that m PQR = r.
Shirlee Remoto Ocampo
Postulates on Angles
Postulate 1.10. Angle Addition
Postulate
If D is in the interior of /_BAC,
then
m/_ BAC = m/_ BAD + m /_DAC.
Shirlee Remoto Ocampo
On Angles
Congruent angles are angles
that have the same measure.
An angle bisector is a ray that
divides an angle into two
congruent angles.
Shirlee Remoto Ocampo
Angle Pairs
Adjacent angles are angles in the
same plane with a common side and
a common vertex but with no
common interior points.
Complementary angles are two
angles whose measures have a sum
of 90o.
Supplementary angles are two
angles whose measures have a sum
of 180o.
Shirlee Remoto Ocampo
Angle Pairs
Linear pair consists of two
angles which are adjacent and
whose noncommon sides are
opposite rays.
Vertical angles are two
nonadjacent angles formed by
two intersecting lines.
Shirlee Remoto Ocampo
Postulate on Angles
Postulate 2.1 Linear Pair
Postulate
If two angles form a linear pair,
then they are supplementary.
Shirlee Remoto Ocampo
Theorems on Angles
Theorem 2.1 If two angles are
supplementary and congruent,
then each is a right angle.
Theorem 2.2 Congruent
Complements Theorem
If two angles are complements
of congruent angles (or of the
same angle) then the two angles
are congruent.
Shirlee Remoto Ocampo
Theorems on Angles
Theorem 2.3 Congruent
Supplements Theorem
If two angles are supplements of
congruent angles (or of the same
angle) then the two angles are
congruent.
Theorem 2.4 Vertical Angles
Theorem
Vertical angles are congruent.
Shirlee Remoto Ocampo
What are polygons?
many-sided figures, with sides
that are line segments
named according to the number
of sides and angles they have
A regular polygon is one that
has equal sides.
Examples of polygons
Non polygons
Sum of interior angels of
polygons
Sum of Interior Angles
of a Polygon
= 180(n - 2)
(where n = number of sides)
Sum of interior angles Start with vertex A and connect it to all
other vertices (it is already connected to B
and E by the sides of the figure). Three
triangles are formed. The sum of the
angles in each triangle contains 180°. The
total number of degrees in all three
triangles will be 3 times
180. Consequently, the sum of the interior
angles of a pentagon is:
3 180 = 540
Notice that a pentagon has 5 sides, and
that 3 triangles were formed by connecting
the vertices. The number of triangles
formed will be 2 less than the number of
sides.
This pattern is constant for all polygons. Representing
the number of sides of a polygon as n, the number of
triangles formed is (n - 2). Since each triangle
contains 180°, the sum of the interior angles of a
polygon is 180(n - 2).
Sum of exterior angles
No matter what type of polygon
you have, the sum of the
exterior angles is ALWAYS
equal to 360°. If you are working
with a regular polygon, you can
determine the size of EACH
exterior angle by simply dividing
the sum, 360, by the number of
angles.
Sum of Exterior angles
An exterior angle of a polygon is an angle that forms a linear pair with one of the angles of the polygon.
Two exterior angles can be formed at each vertex of a polygon. The exterior angle is formed by one side of the polygon and the extension of the adjacent side.
Theorems on Polygons
Theorem 2.5 The number of
diagonals that can be drawn in a
polygon of n-sides is .
Shirlee Remoto Ocampo
Theorem 2.6 Polygon
Interior Angle-Sum Theorem
The sum of the measures of
the interior angles of a convex
polygon of n sides is (n-2)180o.
Corollary: The measure of each
angle of a regular polygon with n
sides is
Shirlee Remoto Ocampo
Theorem 2.7 Polygon Exterior
Angle-Sum Theorem
The sum of the measures of all
exterior angles of a polygon, one
at each vertex, is 360o.
Shirlee Remoto Ocampo
Shirlee Remoto Ocampo
REFERENCE
Remoto-Ocampo, Shirlee.(2010)
Mathematics Ideas and Life
Applications (MILA) III:
Geometry, Philippines: ABIVA
Publishing, Inc.
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