TOOLS OF GEOMETRY - Weebly · 2019. 2. 18. · geometry these are words that do not require a formal definition. They form the building blocks for formally defining or proving other

TOOLS OF GEOMETRY

Shirlee Remoto Ocampo

De La Salle University-Manila


“There is no royal road to Geometry.” - Euclid

Let us take a journey…

G E O M E T R Y


Eye and

brain….vision

and mind

working

together?


Are the lines parallel?


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.


What figure can you see?


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


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.


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’?


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


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


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


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.


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.


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.


Postulates on Angles

Postulate 1.10. Angle Addition

Postulate

If D is in the interior of /_BAC,

then

m/_ BAC = m/_ BAD + m /_DAC.


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.


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.


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.


Postulate on Angles

Postulate 2.1 Linear Pair

Postulate

If two angles form a linear pair,

then they are supplementary.


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.


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.


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 .


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


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.



REFERENCE

Remoto-Ocampo, Shirlee.(2010)

Mathematics Ideas and Life

Applications (MILA) III:

Geometry, Philippines: ABIVA

Publishing, Inc.

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TOOLS OF GEOMETRY - Weebly · 2019. 2. 18. · geometry these are words that do not require a formal definition. They form the building blocks for formally defining or proving other