Mathematics 8 Quarter 3 Module 2: Illustrating Axiomatic Structures of a Mathematical System

Mathematics Quarter 3 – Module 2 Illustrating Axiomatic Structures of a Mathematical System

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Mathematics

Quarter 3 – Module 2

Illustrating Axiomatic Structures of a Mathematical System

[illustration: students learning outdoors and at home around a globe, a lamp, and a tree]

Development Team of the Module

Writers: Alma R. Velasco, Hezl M. Evangelio

Language Editor: Vicente P. Balbuena

Content Evaluator: Margie T. Tambis, Nilbeth S. Merano, Rosalita A. Bastasa, Merjorie G. Dalagan

Layout Evaluator: Jay R. Tinambacan

Reviewers: Rhea J. Yparraguirre, Severiano D. Casil, Crisante D. Cresino

Illustrator: Alma R. Velasco, Hezl M. Evangelio, Marieto Cleben V. Lozada, John Merick L. Cifra

Layout Artist: Alma R. Velasco, Hezl M. Evangelio

Management Team: Francis Cesar B. Bringas, Isidro M. Biol, Jr., Maripaz F. Magno, Josephine Chonie M. Obseñares, Josita B. Carmen, Celsa A. Casa, Regina Euann A. Puerto, Bryan L. Arreo, Elnie Anthony P. Barcena, Leopardo P. Cortes, Jr., Claire Ann P. Gonzaga

Mathematics – Grade 8

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Quarter 3 – Module 2 Illustrating Axiomatic Structures of a mathematical System First Edition, 2020

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8

Mathematics

Quarter 3 – Module 2

Illustrating Axiomatic Structures of a Mathematical System

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

What I Need to Know

[illustration: hand pressing a button inside a circle]

In this module, you will learn the axiomatic structure of a mathematical system and why there is a need to learn them. The scope of this module enables you to use it in many different learning situations. The lesson is arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using.

This module contains:

Lesson 1 – Illustrating Axiomatic Structures of a Mathematical System

Objectives: After going through this module, you are expected to:

1. define axiomatic system;
2. determine the importance of an axiomatic system in geometry;
3. illustrate the undefined terms; and
4. cite definitions, postulates, and theorems involving points, lines and planes.

What I Know

[illustration: hand holding a pencil inside a circle]

Pre-Assessment

Directions: Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper.

1. Which undefined term has a specific position but has no dimension or direction?
   A. point C. plane
   B. line D. angle

   
   

2. Which of the following represents a plane?
   A. tip of a pen C. top of the table
   B. stretched rope D. corner of a rectangular box

   
   

3. What do you call the lines that intersect at a point forming a right angle?
   A. diagonals C. parallel lines
   B. line segments D. perpendicular lines

   
   

4. Two angles with sum equal to 180°.
   A. acute angles C. nonadjacent angles
   B. vertical angles D. supplementary angles

   
   

5. An axiomatic system is a set of what?
   I. Undefined and defined terms IV. Doubts
   II. Guesses V. Axioms
   III. Proof VI. Theorems
   A. I, II, III C. I, II, IV, V
   B. IV, V, VI D. I, III, V, VI

   
   

6. Which of the following statements is true?
   A. An axiomatic system is independent if there are no contradicting axioms or theorems.
   B. Consistency is a necessary requirement for an axiomatic system to be logically valid.
   C. Absolute consistency of the axiomatic system is established if an abstract model has been exhibited.
   D. In an axiomatic system, every statement, either itself or its negation, is capable of being proven true or false.

   
   

7. What undefined term is a one-dimensional figure that has infinite points and extends indefinitely in opposite direction?
   A. Point C. Plane
   B. Line D. Angle

   
   

8. What axiom of equality is represented by the illustration below?

   
   

[illustration: quadrilateral OHTS with points O, S, H, T labeled; a segment OT drawn inside; text beside it reads “Given: TO ≅ TO”]

A. transitive property C. symmetric property B. reflexive property D. substitution property

9. What theorem states that the exterior angle of a triangle is equal to the sum of two remote interior angles of the triangle?
   A. isosceles triangle theorem C. linear pair theorem
   B. exterior angle theorem D. vertical angles theorem

   
   

10. A line which passes through the midpoint of segment at right angles.
    A. diagonal C. congruent segments
    B. line segment D. perpendicular bisector

    
    

11. What figure is formed when two noncollinear rays meet in a common point?
    A. square C. plane
    B. triangle D. angle

    
    

12. Which of the following statements represents the Segment Addition Postulate?
    A. Points A, B, C are collinear and B is between A and B, then $AB + BC = AC$
    B. Points A, B, C are collinear and B is between A and B, then $AB + AC = BC$
    C. Points A, B, C are collinear and B is between A and B, then $AB = AC = BC$
    D. Points A, B, C are collinear and B is between A and B, then $AB – BC = AC$

    
    

13. Which statement justifies congruent angles?
    A. If two lines intersect C. If it has the same measure
    B. If it has a common side D. If the sum of two angles is 180°

    
    

14. A triangular kite LMC has the following interior angles, 50°, 50°, and 80°. Liza concluded that the kite is an isosceles triangle. What theorem would support her conclusion?
    A. isosceles triangle theorem C. linear pair theorem
    B. exterior angle theorem D. vertical angles theorem

    
    

15. You are tasked to make a triangular picture frame and your teacher gives you two sticks of the same length and one shorter length. What type of triangle can you make out of the materials given if you are not allowed to cut the stick?
    A. scalene triangle C. isosceles triangle
    B. equilateral triangles D. the materials cannot form a triangle

    
    

Lesson 1 Illustrating Axiomatic Structures of a Mathematical System

In this module you will learn about the terms in geometry which are said to be the bases in defining other geometric terms and formulating postulates which could be used to derive a logical result. These terms are said to be the building blocks of geometry. The discussion in this module will help you answer the question “Why is there a need for an axiomatic structure in geometry?”

Let us start this module by reviewing the axioms for real numbers.

What’s In

[illustration: hand holding a puzzle piece inside a circle]

Activity: Color me!

Directions: Determine the axiom illustrated by the statement in each of the following polygons. Color the polygon based on the legend given in a box at the right, then answer the questions that follow.

1. [illustration: triangle containing “x + 2 = 2 + x”]
2. [illustration: trapezoid containing “4×1 = 1×4 = 4”]
3. [illustration: rectangle containing “For every real numbers x and y, one of the following is true: x = y, x > y or x < y”]
4. [illustration: diamond containing “(5 + 2) + 3 = 5 + (2 + 3)”]
5. [illustration: hexagon containing “If x + 3 > y and y > z−5, then x + 3 > z − r”]

Color Legend:

Color	Axiom
Red	Trichotomy Axiom
Green	Distributive Axiom
Blue	Transitive Axiom
Yellow	Associative Axiom
Orange	Distributive Axiom
White	Commutative Axiom
Brown	Additon or Multiplication Axiom
Pink	Existence of Additive or Multiplicative Inverse
Violet	Existence of Additive or Multiplicative Identity

[transcriber’s note: the table has no header row in the original; “Color | Axiom” is added as the header for the two printed columns. “Additon” is printed as such.]

Questions:

1. Where you able to recall the axioms for real numbers illustrated in each polygon?
2. What axiom will justify the given statement, 10 + (−10) = 0?
3. Describe your feeling as you color each polygon.
4. Do you know that psychological reactions to color could alter attention, mode, and motivation to learn?
5. Colors have effects on children’s bodily functions, mind and emotions (Renk Etkis, 2017). Blue and green colors are associated to calmness, comfort, happiness and contentment, pink reduces aggressive behavior, while little red and yellow is helpful in drawing children’s attention. What about you? What color do you think would motivate you to learn?

Now that you had picked your color, have it with you so it will get you motivated to learn and will make you ready to perform the next activity.

What’s New

Activity 1: Visualize me!

Directions: Complete the table below by writing the appropriate representation of point, line, and plane found in the box.

Point	Line	Plane

top of the table, edge of the cabinet, tip of a pencil edge of a book, tip of a marker, wall of a classroom stretched rope, edge of a ruler, cellphone screen corner of a rectangular tray

What is It

Robots are increasingly used in many industries in the whole world, from healthcare and manufacturing to defense and education.

For instance, a certain robot is being used in an industry or establishment to do a particular activity. Suppose that the statements below describe the routine for such robot to control activity in a warehouse:

[illustration: a robot]

Set of statements:

Statement 1: Every robot has at least two paths.

Statement 2: Every path has at least two robots.

Statement 3: A minimum of one robot exist.

In this set of statements, which do you think are terms that need to be defined?

Suppose you are asked to prove another statement, say, “a minimum of one robot exists”, can you use these three statements to prove it?

The set of statements above are true and contain terms that are undefined or needs to be defined. These statements can be used to create and to prove another statement. These set of statements are examples of an axiomatic system.

An axiomatic system is a logical system which possesses an explicitly stated set of axioms from which theorems can be derived.

From the definition, you could say that axiomatic system consists of some undefined terms (also called the primitive terms), defined terms, list of axioms or postulates concerning the undefined terms, a system of logic (or proofs) to be used in deducing new statements called theorems.

The axiomatic structure of a mathematical system can be compared to a tree.

The ground is not part of the tree but it is necessary for the tree to be planted and to grow. Like the ground, an axiomatic system needs a logical system of rules that allows one to make inferences.

[chart: diagram; tree with five labeled levels]

Theorems – Branches and Leaves. Axioms – Trunk Definitions – Roots Undefined terms – Root Logical System – Ground Undefined terms – Root Tips

[chart: diagram; circular definition]

collection, set, group Circular definition

The roots at the base of the tree correspond to the undefined and defined terms of the system. These are the basic term from which statements in the axiomatic system are made. Like the tip roots combined to build up roots of the tree, undefined terms are the starting point for every definition and statement of the system. They are combined in various ways into a statement called the definition.

Terms are defined for us to be precise and concise on its meaning. But there are some basic terms of the system that are necessarily left undefined. When we define a term, we will be using different terms that we also need to define. For instance, we want to define the word “set”. Looking up for the word “set” in the dictionary, you will find that “set is a group of objects or numbers.” Then you also have to define the term “group”, which means “collection”, and collection means “set”. The process is a circular definition. Thus, there are basic terms left undefined to prevent circular definition.

The trunk of the tree corresponds to the axioms or postulates while the branches growing out of that trunk are the theorems. Axioms are the statements that serve as a starting point for the system. Axioms are the basic truths and we used them to prove other statements. Theorems, on the other hand, are statements deduced from the axioms.

Axiomatic system has three properties.

1. Consistency.

An axiomatic system is said to be consistent if there are no axiom or theorem that contradict each other. This means that it is impossible to derive both a statement and its negation from the axiom set of system.

Example:

Axiom statement: There exist two lines that are parallel.

Negation: No two lines are parallel.

Notice that the negation is not an axiom nor a theorem.

The system where a statement and its negation are both true is said to be inconsistent.

An axiomatic system should be consistent for it to be logically valid. This means that there are no axiom or theorems that contradict each other. Otherwise, the axiomatic system is faulty or inaccurate.

2. Independence.

In an axiomatic system, an axiom or postulate is said to be independent if it is not a theorem that follows the other axioms. It is not a theorem that can be derived or cannot be proven true using other axioms in the system.

For instance, you have four different axioms. If you can make a model showing that one axiom is independent of the other, that is, you cannot use the other three axioms to prove such axiom, then the axiom is independent.

An example to this is Euclid’s fifth postulate. Many people tried to prove this axiom using the other four postulates but either failed or used faulty reasoning. This problem led to the development of other geometries where the fifth theorem of Euclid was shown to be independent of the other postulates.

Independence is not a necessary requirement for an axiomatic system.

3. Completeness.

An axiomatic system is complete if for every statement, either itself or its negation, is derivable in that system. In other words, every statement is capable of being proven true or false.

Now that you have learned the axiomatic system, let us try to apply it in the situation mentioned earlier about the artificial axiomatic system describing a routine for a computer to control activity of the robot in a warehouse. The set of axioms given were:

Axiom 1. Every robot has at least two paths. Axiom 2. Every path has at least two robots. Axiom 3. There exists at least one robot.

a. What are the undefined terms in this axiom set?

Answer:

In this system, the undefined terms are “robot”, “path” and “has”. The terms “robot” and “path” are elements and the term “has” is a relation. This indicates that there is some relationship between robot and path. These undefined terms can be used to build construct to various proofs.

b. If you are asked to prove say, “Theorem 1. There exists at least one path.”, how would you do it?

Answer:

Notice that Axiom 3 guarantees that a robot exists but no axiom clearly states that there is a path. The sequence of proof could be as follow:

Proof:

1. By the third axiom, there is an existence of a robot.
2. By the first axiom, each robot must have at least two paths.
3. Therefore, there exist at least one path.

Notice that Axiom 3 is a consequence of Axiom 1 and Axiom 2.

c. What is the minimum number of paths? Prove it.

Answer: Notice that Axiom 1 states that every robot has at least two paths. Hence, the minimum number of paths is two.

Proof:

1. By the third axiom, a robot exists, call it R₁.
2. By the first axiom, R₁ must have at least two paths call them P₁ and P₂.
3. Therefore, at least two paths exist.

The example above clearly shows that an axiomatic system is a collection of axioms, or statement about undefined terms, from which proofs and theorems or logical arguments are built.

The following are some examples and illustrations of each part in the axiomatic structure.

Undefined terms

Axiomatic structure started with three undefined terms (or primitive terms): point, line, and plane. These terms are the bases in defining new terms, hence they are called the building blocks of geometry. Even though they are called undefined terms, it does not really mean that we are restricted to describe or represent them.

The table below shows the different ways of describing these three undefined terms.

Point	Line	Plane
Something having specific position but it has no dimension (no length, no width, and no thickness) or direction.	A one-dimensional figure with infinite numbers of points, no specific length, without width nor thickness. It is always straight that extends indefinitely in two opposite directions.	A flat surface where infinite numbers of lines can lie. It has no specific length and width and without thickness. It extends indefinitely in all directions.
There are objects that illustrates point, line, and plane in real-world. In your previous activity, they are:
1. tip of a pencil; 2. tip of a marker; 3. corner of a rectangular tray;	1. edge of the cabinet; 2. edge of a book; 3. stretched rope; 4. edge of a ruler.	1. top of the table; 2. wall of a classroom; 3. cellphone screen;

Now look around you, can you name something in your place that would illustrate point, line, and plane?
A point can be represented with a dot and is denoted by a capital letter. The two points below are point A and point B.	The line below is line AB, denoted by the symbol AB⃡, which is named after the two points that are on the line. Lines can also be denoted by a lower case letter like line l below.	The parallelogram below is a plane denoted by the Greek Letter α, read as ‘alpha’. This plane can also be named as plane ABC.

[illustration: points A and B]

[illustration: line AB with points A and B and line labeled l]

[illustration: plane α with points A, B, and C]

Defined terms

From these three undefined terms, important concepts in geometry will be defined. Remember that we need defined terms because we want to be precise and concise on the meaning of a term. Definitions will enable us to understand each other and to make sure we mean the same thing about a certain term.

Below are some definitions derived from the undefined terms, point, line, and plane.

1. Definition of a Segment

Segment AB, denoted by AB or BA, is the union of points A, B and all the points between them. A and B are called the endpoints of the segments.

[illustration: segment with endpoints A and B]

2. Definition of Between

Point U is said to be between F and N if and only if F, U, and N are distinct points of the same line and FU + UN = FN.

[illustration: line with points F, U, N; 2 cm between F and U, 3 cm between U and N, 2cm + 3cm = 5cm overall]

In FN, U is between F and N since F, U, and N are distinct points on the same line and FU + UN = FN and 2cm + 3cm = 5cm.

3. Definition of Collinear Points and Coplanar Points

When points are on the same line, they are called collinear points.

[illustration: points A, B, C on a line and point D above the line]

Note that points A, B, and C are on the same line, hence they are said to be collinear, while point D is not on the same line with the other three, thus, these four points A, B, C, and D are noncollinear.

When points are on the same plane, they are called coplanar points.

[illustration: plane P with points B, I, G]

Notice that points B, I, and G are on the same plane P, hence they are said to be coplanar.

Can points and lines be coplanar? The answer is yes. As long as they are on the same plane, they are said to be coplanar like the one illustrated below.

[illustration: plane S with points O, J, Y and line n]

In the figure, points J, O, Y and line n are all on the same plane S, hence they are coplanar.

4. Definition of a Ray

Ray is a part of a line that has one endpoint and goes on infinitely in one direction.

[illustration: line with points P, Q, R and an arrow pointing right]

In the figure above, ray PQ starts from point P and goes on to the right without bound. P is called the endpoint of $\overrightarrow{PQ}$. Can you call it ray PR? The answer is yes.

5. Definition of an Angle

An angle is the union of two noncollinear rays with a common endpoint.

[illustration: angle with vertex U, rays UT and UV, angle marked 1]

The common endpoint being shared by $\overrightarrow{UT}$ and $\overrightarrow{UV}$ is point U called the vertex. The angle formed could be named as ∠1 or ∠TUV or ∠VUT.

6. Definition of Congruent Angles

Two angles are congruent if and only if their measures are equal.

In symbol: ∠X ≅ ∠Y, if and only if m∠X = m∠Y.

[illustration: two angles, X and Y, each marked 45°, labeled m∠X = 45° and m∠Y = 45°]

7. Definitions of Acute Angle, Right Angle, and Obtuse Angle

An acute angle is an angle with a measure greater than 0° but less than 90°.

A right angle is an angle with a measure of 90° .

An obtuse angle is an angle with a measure greater than 90° but less than 180°.

[illustration: three angles. Acute Angle: vertex B, rays to A and C, marked 30°. Right Angle: vertex E, rays to D and F, marked 90°. Obtuse Angle: vertex H, rays to G and I, marked 100°]

8. Definition of Adjacent Angles

Adjacent angles share a common vertex and a common side, but do not overlap.

[illustration: angle ABD and angle CBD sharing vertex B and side BD; angle EFG and angle GFH sharing vertex F and side FG]

∠ABD and ∠CBD are adjacent angles which have a common vertex B and a common side $\overline{BD}$. The interiors of ∠ABD and ∠CBD do not intersect and therefore, the two angles have no interior points in common.

∠EFG and ∠GFH have F as common vertex, and $\overline{FG}$ as common side, but the interiors of the two angles intersect, this means that the two angles have common interior points. Thus, ∠EFG and ∠GFH are not adjacent angles.

9. Definition of Supplementary Angles

Two angles are supplementary when the sum of their angles is 180°.

[illustration: straight line with a ray forming angles 1 and 2, labeled m∠1 + m∠2 = 180°]

[illustration: angles 3 and 4 marked 120° and 60°, labeled m∠3 + m∠4 = 120° + 60° = 180°]

10. Definition of Linear Pairs

A linear pair of angles is formed when two lines intersect. Two angles are said to be linear if they are adjacent angles formed by two intersecting lines and are supplementary.

[illustration: line A–B–C with ray BD from point B, forming angles ABD and CBD]

$\overrightarrow{BD}$ is the common side, $\overrightarrow{BA}$ and $\overrightarrow{BC}$ are opposite rays, ∠ABD and ∠CBD forms a linear pair.

11. Definition of Vertical Angles

Opposite angles formed by two intersecting lines are vertical angles.

[illustration: two intersecting lines with the four angles numbered 1, 2, 3, and 4]

∠1 and ∠3 are vertical angles. ∠2 and ∠4 are also vertical angles. Noticed that these angles are opposite each other.

12. Definition of Perpendicular Lines

Perpendicular lines are two lines that intersect to form a right angle.

[illustration: line AB vertical and line CD horizontal, intersecting at E]

$\overleftrightarrow{AB}$ intersects $\overleftrightarrow{CD}$ at point E. ∠AED, ∠AEC, ∠BEC, and ∠BED are right angles form by these two intersecting lines, hence, $\overleftrightarrow{AB}$ is perpendicular to $\overleftrightarrow{CD}$. In symbol, $\overleftrightarrow{AB} \perp \overleftrightarrow{CD}$.

13. Definition of Perpendicular Bisector

A perpendicular bisector $PR$ of a line segment $XZ$ is a line segment perpendicular to $XZ$ and passing through the midpoint.

[illustration: segment XZ with midpoint Y, and line segment PR perpendicular to it at Y, with P above and R below]

$\overline{PR} \perp \overline{XZ}$ at point Y. $\overline{PR}$ divides $\overline{XZ}$ into two equal parts. Thus, $XY = ZY$. Furthermore, $\overline{XY} \cong \overline{ZY}$.

14. Definition of Polygon

A polygon is a closed figure such that the union of three or more coplanar segments, which intersect at endpoints, with each endpoint shared by exactly two noncollinear segments.

[illustration: boxed group labeled “Polygon” showing a triangle, a square, and an L-shaped figure]

[illustration: boxed group labeled “Not Polygon” showing a figure with points A, B, C, D, and E, an open angle, an octagon with rounded corners, and a curved figure]

Point C is shared by more than two segments.

15. Definition of Convex Polygon

A polygon is convex if and only if the lines containing the sides of the polygon do not contain points in its interior.

[illustration: quadrilateral ABCD with dashed lines extended from its sides, labeled Exterior and Interior]

[illustration: quadrilateral EFGH with both diagonals drawn]

If each diagonal, except its endpoints, is entirely in the interior of the polygon, then the polygon is convex, like polygon EFGH.

16. Definition of Nonconvex (Concave) Polygon

A polygon is nonconvex (concave) if and only if at least one of its sides is contained in a line, which contains also points in the interior of the polygon.

[illustration: polygon JKLM with a dashed line extended through side LM]

$\overleftrightarrow{LM}$ which contains $\overline{LM}$ also contains points in the interior of the polygon, hence, polygon JKLM

17. Definition of Regular Polygon

A regular polygon is a polygon that is both equilateral and equiangular.

[illustration: regular pentagon DREAM with center O, labeled Interior angle at vertex R and Central angle at O]

[illustration: regular pentagon DREAM with center O and extended side MN, labeled Exterior angle at vertex M]

The regular polygon DREAM has five interior angles ∠RDM, ∠ERD, ∠AER, ∠EAM, ∠AMD. These angles are equal, m∠RDM = m∠ERD = m∠AER = m∠EAM = m∠AMD.

The polygon DREAM has five equal sides, DR = RE = EA = AM = MD. Point O is the center of the given polygon. ∠O is the central angle.

Regular polygon DREAM has also five exterior angles. These angles are obtained when one of the intersecting sides is extended such as ∠AMN. The outside angle along with the vertex is an exterior angle.

18. Definition of a Triangle

A triangle is a three-sided polygon.

[illustration: triangle with vertices R (top), T (bottom left), and I (bottom right)]

The symbol “∆” followed by the three letters representing the noncollinear points (or the vertices) is used to name the triangle. Every triangle, like ∆TRI has three sides ($\overline{TR}$, $\overline{RI}$, $\overline{TI}$), three angles (∠T, ∠R, ∠I), and three vertices (T, R, I).

19. Definition of Angle Bisector of a Triangle

An angle bisector of a triangle is a segment contained in the ray, which bisects the angle of the triangle, and whose endpoints are the vertex of this angle and a point on the opposite side.

[illustration: triangle LOV with vertex O at the top and bisector OE drawn to side LV at E]

$\overline{OE}$ is an angle bisector of ∆LOV. It is the bisector of ∠LOV. The endpoint O of the angle bisector is the vertex of ∆LOV and the other endpoint E is on the opposite side. Thus, ∠LOE ≅ ∠EOV.

20. Definition of an Altitude of a Triangle

A segment is an altitude of a triangle if and only if it is perpendicular from a vertex of the triangle to the line that contains the opposite side.

[illustration: triangle STA with vertex T and altitude TY to side SA, with a right-angle mark at Y]

Every triangle has three altitudes. In ∆STA, $\overline{TY}$ is one of the three altitudes.

[illustration: triangle SAF with altitude SE drawn from vertex S to the extended line through E and F, with a right-angle mark at E]

In ∆SAF, $\overline{SE}$ is one of the three altitudes.

[illustration: triangle LSE with altitudes LI, SN, and TE, each with a right-angle mark]

In ∆LSE, $\overline{LI}$, $\overline{SN}$, and $\overline{TE}$ are the three altitudes.

21. Definition of Median

A segment is a median of a triangle if and only if its endpoints are a vertex and the midpoint of the opposite side.

[illustration: triangle with vertices R (top), N (bottom left), and A (bottom right), with midpoints O, M, and L marked by tick marks, and three medians drawn]

Every triangle has also three medians. Median, except its endpoints, is always in the triangle’s interior. Unlike altitude that can be drawn from the exterior of the triangle. $\overline{AO}$ is the median to $\overline{NR}$, $\overline{RL}$ is the median to $\overline{NA}$, and $\overline{NM}$ is the median to $\overline{RA}$.

22. Definitions of Acute, Right, Obtuse, and Equiangular Triangle

An acute triangle is a triangle in which all angles are acute.

A right triangle is a triangle in which one of the angles is a right angle.

An obtuse triangle is a triangle in which one of the angles is obtuse.

An equiangular triangle is a triangle in which all angles are congruent.

[illustration: Acute Triangle with vertices O, H, and T]

[illustration: Right Triangle with vertices D, R, and Y, with a right-angle mark at R]

[illustration: Obtuse Triangle with vertices S, U, and N]

[illustration: Equiangular Triangle with vertices D, Y, and A, with angle marks]

23. Definitions of Scalene, Isosceles, and Equilateral Triangle

A scalene triangle is a triangle with no congruent sides.

An isosceles triangle is a triangle with at least two congruent sides.

An equilateral triangle is a triangle with all sides congruent.

[illustration: Scalene Triangle with vertices H, R, and E]

[illustration: Isosceles Triangle with vertices B, G, and I, with two sides marked congruent. $\overline{BG} \cong \overline{BI}$]

[illustration: Equilateral Triangle with vertices A, H, and T, with all sides marked congruent. $\overline{HA} \cong \overline{AT} \cong \overline{HT}$]

Take note of these definitions because you will be using them in your future lessons.

Axioms

In the axiomatic structure of a mathematical system, axiom is defined as a logical statement accepted to be true without proof. Axioms can be used as a premise in a deductive argument. In the Elements, Euclid presented 10 assumptions, five of which are not specific to geometry, and he called them common notions (axioms), while the other five are specifically geometric in which he called them postulates.

The following are examples:

Common Notions (Axioms):

Axiom 1. Things which are equal to the same thing are also equal to one another. This is transitive property of equality.

[illustration: three diamonds labeled A, B, and C with equal signs between them]

If A = B and B = C, then A = C.

[illustration: segment AB = 12 cm and segment CD = AB on a ruler]

Since $\overline{AB} = 12\ cm$ and $\overline{CD} = \overline{AB}$, then $\overline{CD} = 12\ cm$.

Axiom 2. If equals are added to equals, the wholes are equal. This is addition property of equality.

[illustration: parallelograms labeled A, B, and C with plus signs, showing A = B and A + C = B + C]

If A = B and C = C, then A + C = B + C.

[illustration: segments AB = 4 cm and DE = 4 cm; BC = 3 cm and EF = 3 cm; AC = 4 cm + 3 cm = 7 cm and DF = 4 cm + 3 cm = 7 cm]

Since $\overline{AB} = \overline{DE}$ and $\overline{BC} = \overline{EF}$, and if $\overline{AB} + \overline{BC} = \overline{DE} + \overline{EF}$, then $\overline{AC} = \overline{DF}$, that is 4cm + 3cm = 4cm + 3cm; 7cm = 7cm.

Axiom 3. If equals are subtracted from equals, the remainders are equal. This is subtraction property of equality.

[illustration: shapes labeled A, B, and C showing A = B and A − C = B − C]

If A = B and C = C, then A − C = B − C.

[illustration: segments AB = 9 cm and DE = 9 cm; BC = 4 cm and FE = 4 cm; AC = 9 cm − 4 cm = 5 cm and DF = 9 cm − 4 cm = 5 cm]

Since $\overline{AB} = \overline{DE}$ and $\overline{CB} = \overline{FE}$, and if $\overline{AB} – \overline{CB} = \overline{DE} – \overline{FE}$, then $\overline{AC} = \overline{DF}$, that is 9cm − 4cm = 9cm − 4cm; 5cm = 5cm.

Axiom 4. Things which coincide with one another are equal to one another. This is reflexive property.

[illustration: triangle a placed over triangle b, showing that the two triangles coincide and are equal]

If the triangle in b fits to the triangle in a, then the triangle in a is equal to the triangle in b.

[illustration: triangle with vertices O, H, R, and C, with segment OR shared; labeled OR ≅ OR, then ∠R = OR [?]]

Any segments, angles and polygons are always equal to themselves.

Axiom 5. The whole is greater than the part.

[illustration: a hamburger shown to be greater than a slice of cake, with the “>” sign between them]

The following axioms below are both used in geometry and other field of mathematics.

• Symmetric Property of Equality

  
  

  ➢ For all real numbers p and q, if p = q, then q = p.

  
  

  [illustration: box labeled P = Q; then box labeled Q = P]

  
  

• Substitution Property of Equality

  
  

  ➢ For all real numbers p and q, if p = q, then q can be substituted for p in any expression.

  
  

  [illustration: angles p and r marked congruent (If ≅ and then); a line with angle q supplementary to angle p; then angle q and angle r]

  
  

  q is supplementary to p

  
  

Geometrical Postulates:

Postulate 1. A straight line segment can be drawn joining any two distinct points.

[illustration: points A and B, with a line segment AB drawn between them]

Given two points A and B, a line segment AB can be drawn.

Postulate 2. Any straight line segment can be extended indefinitely in a straight line.

[illustration: line segment CD, and the same line extended in both directions with an arrow at each end]

Line segment CD can be extended indefinitely in both direction.

Postulate 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

[illustration: segment CD and a circle centered at C with CD as its radius]

Point C is the center of the circle.

Line segment CD is the radius of circle C.

What’s More

Activity 1: Make me meaningful!

Directions: The following is an axiomatic system. Answer each question as required.

Axiom Set:

Axiom 1: Each line is a set of three points. Axiom 2: Each point is contained by two lines. Axiom 3: Two distinct lines intersect at exactly one point.

Questions:

1. What are the undefined terms in this axiom set?
2. Is the axiomatic system consistent? Why? Why Not? State what specific property is the given axiomatic system.

Activity 2: “Who am I”

Directions: Write the definition, postulate, or theorem that supports each statement.

Illustration	Statement	What Definition, Postulate or theorem
Example [illustration: triangle with angles labeled o, h, and t, and one side extended beyond t]	$m∠t = m∠h + m∠o$	Exterior Angle Theorem
1. [illustration: points A, B, and C on a line]	$AB + BC = AC$
2. [illustration: point A with rays AR, AC, and AE]	$m∠RAC + CAE = m∠RAE$
3. [illustration: line PY with ray NO above point N]	$m∠PNO + m∠ONY$

Activity 3: “Write the missing reasons”

Directions: Using the figure below write the missing reasons corresponding to its statement. Number six is done for you.

Given: $m∠MAP = m∠CAR$ Prove: $m∠1 = m∠3$

[illustration: rays AM, AC, AP, and AR from point A, with angles labeled 1, 2, and 3]

transcriber’s note: the word “Figure” printed beneath the word “Statement” in the original overlaps the table header; the header is transcribed as “Statement” and “Reason”.

Statement	Reason
1. $m∠MAP = m∠CAR$	1.
2. $m∠MAP = m∠1 + m∠2$	2.
3. $m∠CAR = m∠2 + m∠3$	3.
4. $m∠1 + m∠2 = m∠2 + m∠3$	4.
5. $m∠2 = m∠2$	5.
6. $m∠1 = m∠3$	6. Subtraction Property

Activity 4: “Identify and Illustrate”

Directions: Given the properties, identify whether it is a theorem or postulate and illustrate.

Properties	Theorem or Postulate	Illustration
1. It states that two angles form a linear pair are supplementary. ➢ $∠PNO$ and $∠YNO$ form a linear pair
2. It states that angles opposite to each other and formed by two intersecting straight lines are congruent. ➢ Line m intersect line n ➢ $∠1$ and $∠3$, $∠2$ and $∠4$ are congruent
3. It states that points $L, M$ and $N$ are collinear and $M$ is between points $L$ and $N$, then $LM + MN = LN$.

What I Have Learned

After going through with this module, it’s now time to check what you have learned from the activities. Read carefully and answer the items that follow.

Directions: Tell whether the following statement is true or false.

_____ 1. An axiomatic system consists of undefined terms, defined terms, axioms, and theorems.

_____ 2. Theorems are proved using undefined terms, defined terms, axioms, a logical system, and/or previous theorems.

_____ 3. A model of an axiomatic system is obtained by assigning meaning to the undefined terms of the axiomatic system in such a way that the axioms are true statements about the assigned concepts.

_____ 4. Euclid’s postulates 1, 2, 3, and 4 can be used to prove postulate 5 (or Euclidean Parallel Postulate).

_____ 5. An axiomatic system should be consistent for it to be logically valid.

_____ 6. Independence and completeness are necessary requirements of an axiomatic system.

_____ 7. Some terms remain undefined to avoid circular definition.

_____ 8. Points, lines and planes are undefined terms in geometry.

_____ 9. Euclid’s first five assumptions are not specific in geometry in which he called them postulates and the other five are specifically geometric which he called them common notions.

_____ 10. In our daily lives, we use true statements based on facts as our reasons in order to arrive at a conclusion. In this manner, we are applying axiomatic system.

What I Can Do

Let’s get this real!

At this point, you will be given a practical task which will demonstrate your understanding on axiomatic system.

Directions: Read the situation below and answer the questions that follow.

Situation: Teacher Clarenda wanted to group here students into committees. She is grouping them according to the following axiom set:

Axiom 1: Each committee consists of exactly two members. Axiom 2: There are exactly six committees. Axiom 3: Each member serves on exactly two committees.

Questions:

1. What are the undefined terms in the axiom set?
2. Let a dot represent a member, and a line represent a committee, make at least one concrete model of the axiom set.
3. Deduce the theorem that “There is a finite number of members.”

Assessment

Directions: Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper.

1. What undefined term is represented by a dot?
   A. point
   B. line
   C. plane
   D. angle

   
   

2. What undefined term is represented by the edge of a ruler?
   A. point
   B. line
   C. plane
   D. angle

   
   

3. Which of the following does not represent a plane?
   A. tip of a pen
   B. top of the table
   C. cover page of a book
   D. faces of a rectangular box

   
   

4. A consistent axiomatic system does NOT do what to itself?
   A. doubt
   B. guess
   C. affirm
   D. contradict

   
   

5. Which of the following statements best describe a perpendicular line?

   • A. a straight line
   • B. intersecting lines
   • C. intersecting lines forming a linear pair
   • D. intersecting lines forming 4 right angles

6. How do you call a line perpendicular to a given segment which divides it into two equal line segments?

   • A. diagonals
   • B. line segment
   • C. congruent segments
   • D. perpendicular bisector

7. What axioms of equality do the illustration below represent?

   Given: EM ≅ EM

   [illustration: triangle with vertices E (top), H (bottom left), and S (bottom right), with segment EM drawn from E to M, where M is on side HS]

   • A. reflexive property
   • B. substitution property
   • C. symmetric property
   • D. transitive property

8. Which of the following represents a plane?

   • A. cover page of ADM module
   • B. edge of the of the book
   • C. corner of a room
   • D. door knob

9. What axioms of equality stated that for all real number 𝑝 and 𝑞, if 𝑝 = 𝑞 then 𝑞 = 𝑝?

   • A. reflexive property
   • B. substitution property
   • C. symmetric property
   • D. transitive property

10. What is the sum of all interior angles of a triangle?

    
    

• A. 90°
• B. 160°
• C. 180°
• D. 360°

11. Which of the statements justify a supplementary angle?

• A. It has a common side.
• B. It has the same measures.
• C. The sum of two angles is 180°.
• D. Two lines intersect.

For items 12 to 13, consider the following axiom set below:

Axiom 1: There are at least two buildings on a school campus. Axiom 2: There is exactly one sidewalk between any two buildings. Axiom 3: Not all the buildings have the same sidewalk between them.

12. What are the undefined terms in given axiom set?

• A. building only
• B. sidewalk only
• C. building and campus
• D. building and sidewalk

13. Arrange the sequence of proof to prove that “there are at least two sidewalks on the school campus.”

I. By Axiom 3, not all buildings have the same sidewalk between them, hence, there must be another building b3 which does not have S12 between it and b1 or b2. II. By axiom 1, there are two buildings denoted by b1 and b2. III. With Axiom 2, there must be a sidewalk between either b1 and b3 or b2 and b3 which is not S12. IV. By axiom 2, there is exactly one sidewalk between b1 and b2 named S12.

• A. I, II, III, IV
• B. II, IV, I, III
• C. III, II, I, IV
• D. IV, III, II, I

14. The angles inside the triangular garden ABC are all 60°. To be able to find the length of the sides of the garden, Mary measured the sides as follows; 2.5 𝑚, 3 𝑚 and 5 𝑚. Did Mary measure it correctly?

• A. No, because an equiangular triangle is also equilateral.
• B. No, because two sides of an equiangular triangle must be equal.
• C. Yes, because a triangle is not equilateral if it is equiangular.
• D. Yes, because not all equiangular triangles are equilateral.

15. You are tasked to make a triangular picture frame and your teacher gives you five sticks of the same length and one shorter length. What triangle can you make out of the materials given if you are not allowed to cut the stick?

• A. equilateral triangles
• B. scalene and isosceles triangles
• C. isosceles and equilateral triangles
• D. The materials cannot form a triangle

Additional Activities

Directions: Complete the table below with the facts associated with the given problems.

Facts	Application/ Illustration	Solution
Vertical angles are congruent.	Use the facts that linear pair forms supplementary angles to prove that vertical angles are congruent. [illustration: two intersecting lines labeled m and l, with the four angles labeled w, x, y, and z]	1. 2. 3. 4. 5. 6. 7. 8. 9.

Answer Key

[transcriber’s note: this page is printed rotated 180 degrees; the answers below were read from the rotated printing and may contain small errors in the layout of the original table]

What I Know

1. A
2. C
3. D
4. D
5. D
6. B
7. B
8. B
9. B
10. D
11. D
12. A
13. C
14. A
15. C

What I Have Learned

1. T
2. F – using only axioms, a logical system, and previous theorems
3. T
4. F – cannot be used
5. T
6. F – are not necessary
7. T
8. T
9. F – common notions are none geometric, are postulates geometric
10. T

What’s In

1. White color (Commutative)
2. Violet (Existence of Multiplicative Identity)
3. Red (Trichotomy Axiom)
4. Yellow (Associative Axiom)
5. Blue (Transitive Axiom)

What’s New

Points

• Corner of a rectangular tray
• Tip of a marker
• Tip of a pencil

Lines

• Stretched rope
• Edge of the cabinet
• Edge of a ruler
• Edge of a book

Plane

• Top of the table
• Wall of a classroom
• Cellphone screen

What I Can Do

1. committee and member
2. Possible model
   [illustration: hexagon with a dot at each vertex]
3. By axiom 1, a member exist.
   By axiom 2, there are exactly 6 committees.
   Therefore, there is a finite number of members.

Assessment

1. A
2. B
3. A
4. D
5. D
6. D
7. A
8. C
9. C
10. C
11. C
12. D
13. B
14. A
15. C

What’s More

Activity 1: Make me meaningful!

1. line and point
2. a.

Axioms	Model 1	Model 2
1	✓	✓
2	✓	✓
3	✓	✓

b. The two models are non-isomorphic because they are not of equivalent structure or they do not look exactly the same. c. The axiomatic system is consistent because the two models are non-isomorphic making the system non-categorical, hence, it is a consistent system.

Activity 2

1. Segment Addition Postulate
2. Angle Addition Postulate
3. Definition of Supplementary angles

Activity 3

1. Given
2. Angle addition Postulate
3. Angle addition Postulate
4. Substitution property
5. Reflexive property

Activity 4

[illustration: three small figures labeled Postulate, Theorem 1, and Theorem 2, each showing lines and angles]

Additional Activities

1. m∠w + m∠x = 180°
2. m∠y + m∠z = 180°
3. m∠w + m∠x = m∠y + m∠z
4. m∠w = m∠y
5. m∠x + m∠y = 180°
6. m∠z + m∠y = 180°
7. m∠x + m∠y = m∠z + m∠y
8. m∠x + m∠y − m∠y = m∠z + m∠y − m∠y
9. m∠x = m∠z

References

Baccay A. (n.d.). Geometry for Secondary Schools. Philippines: Phoenix Publishing House

Burns, C (2011, March 18). More on exterior Angles in triangles. Retrieved August 20, 2020 from http://onemathematicalcat.org/Math/Geometry_obj/exterior_angles.htm

Dailymotion. (2005-2012). Real life application of triangle inequality theorem. Retrieved August 6, 2020, from http://www.dailymotion.com/video/xgeb9n_how-to-apply-the-triangleinequality-theorem-to-real-life-problems_tech

Dhanalekshmi p s, Bed Mathematics

Jurgensen, R. J. Brown, and J.W. Jurgensen (1990). Mathematics 2 An Integrated Approach. Quezon City: Abiva Publishing Huse, Inc.

Moise, E. and F. Downs, Jr. (1997). Geometry Metric Edition. Philippines: Addison- Wesley Publishing Company, Inc.

Renk Etkisi. (2017). The effects of colors on children. Retrieved August 6, 2020, from http://renketkisi.com/en/the-effects-of-colors-on-children.html

Romero, Karl Freidrich Jose D.. Geometr in the Real World. Antonio Arnaiz cor. Chino Roces Avenues, Makati City

Strader, W. and L. Rhoads (1934). Plane Geometry, Philippine Islands: The john C. Winston Company.

Yeo, J., Yee, C., Meng, C., Seng, T.K., Chow, I., and Hong, O.C.(2016). New Syllabus Mathematics. Philippines: Rex Bookstore, Inc.

Websites:

http://web.mnstate.edu/peil/geometry/C1AxiomSystem/AxSysWorksheet.ht

http://www.ms.uky.edu/~lee/ma341/chap1.pdf

http://web.mnstate.edu/jamesju/Spr2010/Content/M487Exam2PracSoln.pdf

https://www.newworldencyclopedia.org/entry/axiomatic_systems

https://www.math.upenn.edu/~mlazar/math170/notes05-2.pdf

For inquiries or feedback, please write or call:

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