DEPARTMENT OF EDUCATION

DIVISION OF SAN JOSE DEL MONTE CITY

San Ignacio St., Poblacion, City of San Jose del Monte, Bulacan 3023
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Mathematics
Quarter III – Module 1
Illustrating and Describing
Mathematical Systems with
Axiomatic Structure
 What I Need to Know

CONTENT STANDARD
The learner demonstrates understanding of key concepts of axiomatic structure of
geometry and triangle congruence.

PERFORMANCE STANDARD
The learner is able to formulate and organize plan to handle a real - life situation and
communicate mathematical thinking with coherence and clarity and in formulating,
investigating, analyzing and solving real life problems involving congruent triangles using
appropriate and accurate representations.

LEARNING COMPETENCY:
The learner describes a mathematical system and illustrate the need for an axiomatic
structure of a mathematical system in general, and in Geometry in particular: (a)defined
terms;
(b) undefined term; (c) postulates; and (d) theorems.

This module was designed and written to help you master the skills in describing and
illustrating mathematical system following axiomatic structure. The scope of this module can
be used in many different learning situations. Throughout this module, you will be provided
with varied activities to process your knowledge and skills acquired. Activities in this module
are arranged accordingly to correspond with your learning needs.
This module contains:
Lesson 1 : Describing Mathematical System
Lesson 2 : Illustrating Axiomatic System

After going through this module, you are expected to:

1. describe mathematical system;
2. identify the different axiomatic structure of a mathematical system; defined
terms, undefined terms, axioms and postulates and theorems;
3. illustrate mathematical systems, and
4. illustrate a mathematical system following axiomatic

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LESSON

1 Describing Mathematical System

God has given us bountiful nature with different shapes and measurements that are
connected with Geometry. Shapes like circles, rectangles, squares and triangles are some
examples, that can be found in our cars, roads, bridges and buildings or other resources that
are available around us. To be able to know, understand and value these, it is important to
have basic knowledge and concepts of Geometry.
In this lesson, you will enumerate the components of mathematical system and begin
to describe them as you move further with the lesson.

What is It
Geometry is the measurement of earthly objects. It started as a way of calculating land
measurement particularly in Egypt and Babylon, where it got its name “geo” which means
earth and “metrein” which means to measure. But Euclid gave geometry a new definition and
developed it into a formal study of statements that involve reasoning. He then defined
geometry as the study of a body of logically connected statements. This makes completely
clear that in geometry, a statement leads to another statement in an arrangement that is
supported by reasons.
In the mathematical system designed by Euclid, there are four parts namely:
1. Undefined Terms
2. Defined Terms
3. Axioms and postulates
4. Theorems

Undefined Terms
There are terms which cannot be defined but can be described. These terms are
called Undefined Terms. The undefined terms in geometry are points, lines and planes
which are said to be the building blocks of geometry. How will you describe these
undefined
terms? Examine the following description and real-life examples below.
Real-life
Term Description Representation Naming
Example

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 A point is
named by a Tip of a pen
A point is a
capital letter.
figure with no
So, the point
Point length and width
below is read as
or simply with
point B.
no dimension.

A line is
determined by
locating two Crease of a
points on the paper
line and it is
named with a
lower-case letter
A line has no or two
width and no capital letters
thickness but it with a double
can be arrowhead
Line above it. The
extended
infinitely in figure below is
opposite read as line BC
directions. or line m and
denoted as ⃡AB

A B

A plane is
named with a
capital letter. Cover of a book
A plane is a flat The plane
surface that below is named
Plane extends as plane N.
infinitely in all
directions.

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Defined Terms
In geometry, defined terms are terms that have a proper definition and can be
defined by means of other geometrical terms. The basic defined terms in geometry are
line segments, rays, opposite rays and angles.
Real-life
Term Description Representation Naming Example
A line segment
is determined
by locating two
Ruler
points and it is
named with a
A line segment
two capital
is formed when
letters. The
two distinct
Line Segment figure below is
points are
read as line
connected with
segment AB
a straight line
and denoted as
̅ 𝐴̅𝐵̅ or ̅𝐵̅𝐴̅ .

A ray is
determined by
locating two
points and it is Arrow
A ray is part of named with two
a line. It capital letters
extends with a single
infinitely in one arrowhead
Ray
direction only above it. The
and stopped by figure below is
a point on one read as ray AB
end. and denoted as
AB.

Angles are
denoted by a
number, vertex Hands of a clock
and the three
capital letters.
An angle is a
The angle
figure formed
below can be
by two rays
Angles named as
meeting at a
∠𝑋𝑂𝑌 or ∠𝑂 or
common
∠1.
endpoint.

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Here are some common terms that you may encounter along with the defined terms;
Term Description Representation Explanation
Midpoint of a is a point that In the figure, we let N be a
segment divides the point on
segment into two ̅𝐴̅
congruent ̅𝐵̅ . Point N is called the
segments ̅𝐴̅𝐵̅,if ̅𝐴̅𝑁̅ is
midpoint of
congruent to 𝑁 ̅ 𝐵
̅ ̅. Congruent
means having equal measure
and similar shape. To denote
congruency, we used “≅”.
Therefore, we can write
̅𝐴̅
̅𝑁̅ ≅
̅ 𝑁̅𝐵̅.
Congruent angles having Since the two right angles
angles equal measure have a measure of
90⁰, therefore ∠𝐴𝑋𝐵 ≅ ∠𝐶𝑌𝐷.

Bisector of an a ray that divides If point Y lies in the interior of

angle an angle into two ∠𝐵𝐶𝐷 and ∠𝐵𝐶𝑌 ≅ ∠𝑌𝐶𝐷 ,
congruent or equal then ∠𝐵𝐶𝐷 is bisected by
angles ⃗𝐶⃗ ⃗
⃗𝑌⃗ , and
⃗⃗𝐶⃗
⃗𝑌⃗ is called the bisector of
∠𝐵𝐶𝐷.
Perpendicular formed when two In the figure, 𝐴 ̅ 𝐶
̅ ̅and ̅𝐵̅𝐷̅
lines lines intersect with intersect at point F to form
each other and right angles. To denote
formed right perpendicularity, we used
angles. The “⊥”. Therefore, we say that
perpendicular sign ̅𝐴̅𝐶̅⊥
is ⊥, it is used to ̅ 𝐵̅
show that two lines
are perpendicular ̅𝐷̅ . Since ∠𝐴𝐹𝐵 and ∠𝐶𝐹𝐵
formed a linear pair, then they
are supplementary. Since
perpendicular lines formed
four right angles, therefore
∠𝐴𝐹𝐵, ∠𝐶𝐹𝐵, ∠𝐶𝐹𝐷 𝑎𝑛𝑑 ∠𝐴𝐹𝐷
measure 90⁰.
Properties of Equality
Reflexive A number is equal
Property to itself (that is, a = By reflexive property, ̅𝐵𝑋
̅ ̅≅
a) ̅ 𝐵̅𝑋̅.

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Symmetric If a = b, then b = a By symmetric property, if
Property ∠𝐴𝐵𝐶 ≅ ∠𝐶𝐷𝐸, then ∠𝐶𝐷𝐸 ≅
∠𝐴𝐵𝐶.

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Transitive If a = b and b = c, By transitive property, if
Property then a = c ̅𝐴̅
̅𝐵̅ ≅
̅ 𝐶̅𝐷̅ and ̅𝐶̅𝐷̅ ≅ ̅𝐸̅𝐹̅ , then ̅𝐴̅𝐵̅
≅
̅ 𝐸𝐹̅ ̅
Axioms and Postulates
Axioms and postulates are both statements that are assumed to be true without any
proof. Their only difference is that, axioms are used in other areas of mathematics while
postulates are widely used in geometry.
Listed below are some of the postulates which are used as guiding rules or
assumptions from which other statements on the undefined terms can be derived.
Remember that these statements are already accepted as true statements.
The Line Postulates
There are two points contained in exactly one line.
Postulate 1

A line contains infinitely many points.

Postulate 2

If two points of a line lie in a plane, then the line lies in the same
plane.
Postulate 3

The Plane Postulates

Any three points lie in at least one

plane, and any three noncollinear
Postulate 4
points lie in exactly one plane.

If two different planes intersect, their

Postulate 5 intersection is a line. In the figure below,
plane E and plane G intersect at line CD.

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 Parallel Postulates
Given a line and a point not on m
the line, there is exactly one line
Postulate 8
through the point parallel to the n
given line. Therefore, m // n.
If two parallel lines are
cut by a transversal,
then the corresponding
Postulate 9 angles are congruent.

In the figure, ∠1 ≅ ∠5
∠2 ≅ ∠6, ∠4 ≅ ∠8, 𝑎𝑛𝑑
∠3 ≅ ∠7.
Theorems
A theorem is a basic geometric principle which are proven to be true by making
connections between accepted definitions, postulates, mathematical operations, and
previously proven theorems.
Since
∠𝐴𝑋𝐵 𝑎𝑛𝑑 ∠𝐶𝑌𝐷
All right angles are are both right
congruent angles, therefore
.
∠𝐴𝑋𝐵 ≅ ∠𝐶𝑌𝐷

Vertical angles are In the figure, ∠𝐴𝐵𝐶 ≅ ∠𝐹𝐵𝐷 and

congruent. ∠𝐴𝐵𝐷 ≅ ∠𝐹𝐵𝐶.

If two parallel lines are

cut by a transversal, Alternate interior angles:
then the alternate ∠3 ≅ ∠5 𝑎𝑛𝑑 ∠4 ≅ ∠6
interior, and alternate Alternate exterior angles:
exterior angles are
∠1 ≅ ∠7 𝑎𝑛𝑑 ∠2 ≅ ∠8
congruent.

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 If two parallel lines are cut by a
transversal, then the interior
angles on the same side of the In the figure, the pairs of interior angles on the same side of the
transversal are transversal are ∠3 𝑎𝑛𝑑∠6, ∠4 𝑎𝑛𝑑 ∠5 .
supplementary. Therefore, 𝑚∠3 + 𝑚∠6 = 180 𝑎𝑛𝑑 𝑚∠4 + 𝑚∠5 = 1800
0

If two parallel lines are cut by a

transversal, then the exterior
In the figure, the pairs of exterior angles on the same side of the
angles on the same side of the
transversal are ∠1 𝑎𝑛𝑑∠8, ∠2 𝑎𝑛𝑑 ∠7 .
transversal are
Therefore, 𝑚∠1 + 𝑚∠8 = 180 𝑎𝑛𝑑 𝑚∠2 + 𝑚∠7 = 1800
0
supplementary.

LESSON

2 Illustrating Axiomatic System

Everything around us involves dimensions, space, shapes and all other things related to
Geometry. It is very important for us to know how things fit together.
In the previous lesson of this module, Mathematical System was defined and described,
which mainly has four parts or axiomatic structure; undefined terms, defined terms, axioms and
postulates, and theorems. In this lesson, illustrative examples of how to illustrate these
mathematical systems through axiomatic structure will be given more emphasis.

What is It

An axiomatic system is a way to establish the mathematical truth that flows from a fixed
set of assumptions. An axiomatic system is a collection of axioms, or statements about undefined
terms. You can build proofs and theorems from axioms. Logical arguments are built from with
axioms.
An axiomatic system needs mathematical systems such as; Undefined terms, Defined
terms, Axioms or Postulates, and Theorems.
The following properties of an axiomatic system should be considered to establish
mathematical truth

Consistency – A statement is said to be consistent if there are no axioms or

theorems that contradict each other

Independence – An axiom is called independent if it cannot be proved or disproved

from the other axioms of the axiomatic system. An axiomatic
system is said to be independent if each of its axioms is
independent.
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 Completeness – An axiomatic system is called complete if every statement
expressible in the terms of the system is either provable or has a
provable negation

Proof is a logical argument in which each statement is supported/justified by given

information, definitions, axioms, postulates, theorems and previously proven statements.

Consider the following examples below.

Example 1
Axiom 1. Every computer set has at least two players.
Axiom 2. Every player has at least two computer set.
Axiom 3. There exist at least one computer set.

Explanation:
This might describe a routine for a shop owner to control activity in a computer shop, but it is also a set of axioms. We have
Llellt's prove a player exist
By the third axiom, a computer set exist.
By the first axiom, the existing computer set must have at least one player. Therefore, at least one player for a computer set

This limited axiomatic system would be enough to build a network of computers to work in a computer shop.

Example 2
Axiom 1. Every line is an intersection of two planes.
Axiom 2. The plane has at least two lines.
Axiom 3. A minimum of one plane exist.

Explanation
Let's prove a plane exist
By the third axiom, a plane exist.
By the first axiom, the line intersect two planes.
By the second axiom, the plane contains at least two lines.

Therefore, if two lines intersect then exactly one plane contains both lines. This prove the Theorem that states “

Example 3
Axiom 1. There are four real numbers.
Axiom 2. The sum of two numbers is equal to the sum of another two numbers.
Axiom 3. At least two numbers are equal
Explanation
By the third axiom, two numbers are equal.
By the first axiom, there are four real numbers.
By the second axiom, the sum of two numbers is equal to the sum of the other two numbers.

Therefore for all real numbers a, b, c and d, if a = b and c = d, then a + c = b + d,

this proves the Addition Property of Equality
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