MATH8-Q3-MODULE2 PDF

Mathematics Quarter 3 – Module 2 Illustrating Axiomatic Structures of a Mathematical System CO_Q3_Mathematics 8_ Module 2  Mathematics – Grade 8 Alternative Delivery Mode Quarter 3 – Module 2 Illustrating Axiomatic Structures of a mathematical System F...

Mathematics Quarter 3 – Module 2 Illustrating Axiomatic Structures of a Mathematical System CO_Q3_Mathematics 8_ Module 2  Mathematics – Grade 8 Alternative Delivery Mode Quarter 3 – Module 2 Illustrating Axiomatic Structures of a mathematical System First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio 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 Printed in the Philippines by ________________________ Department of Education – Caraga Region Office Address: Learning Resource Management Section (LRMS) J.P. Rosales Avenue, Butuan City, Philippines 8600 Telefax Nos.: (085) 342-8207 / (085) 342-5969 E-mail Address: [email protected]  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 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. 1 CO_Q3_Mathematics 8_ Module 2  What I Know 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 2 CO_Q3_Mathematics 8_ Module 2 8. What axiom of equality is represented by the illustration below? O S Given: 𝑇𝑂 ≅ 𝑇𝑂 H T 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 3 CO_Q3_Mathematics 8_ Module 2 Lesson Illustrating Axiomatic Structures of a 1 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 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. 4 CO_Q3_Mathematics 8_ Module 2 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 5 CO_Q3_Mathematics 8_ Module 2  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: 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. 6 CO_Q3_Mathematics 8_ Module 2  Theorems - Branches and Leaves. Axioms - Trunk Definitions - Roots Undefined terms - Root Tips Logical System - Ground 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 collection set 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”. group Looking up for the word “set” in the Circular definition 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. 7 CO_Q3_Mathematics 8_ Module 2 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. 8 CO_Q3_Mathematics 8_ Module 2 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. 9 CO_Q3_Mathematics 8_ Module 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 A one-dimensional A flat surface where specific position but it figure with infinite infinite numbers of lines has no dimension (no numbers of points, no can lie. It has no specific length, no width, and specific length, without length and width and no thickness) or width nor thickness. It is without thickness. It direction. always straight that extends indefinitely in all extends indefinitely in directions. two opposite directions. There are objects that illustrates point, line, and plane in real-world. In your previous activity, they are: 1. tip of a pencil; 1. edge of the cabinet; 1. top of the table; 2. tip of a marker; 2. edge of a book; 2. wall of a classroom; 3. corner of a 3. stretched rope; 3. cellphone screen; rectangular tray; 4. edge of a ruler. 10 CO_Q3_Mathematics 8_ Module 2  Now look around you, can you name something in your place that would illustrate point, line, and plane? A point can be The line below is The parallelogram represented with a dot line AB, denoted by the below is a plane denoted and is denoted by a symbol ⃡𝐴𝐵, which is by the Greek Letter 𝛼, capital letter. named after the two read as ‘alpha’. This The two points points that are on the plane can also be below are point A and line. named as plane ABC. point B. Lines can also be denoted by a lower case letter like line 𝑙 below. A 𝛼 A B B C 𝑙 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. A B 11 CO_Q3_Mathematics 8_ Module 2 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. 2 cm 3 cm ⃡ , 𝑼 is between 𝑭 and 𝑵 In 𝐹𝑁 since 𝐹, 𝑈, and 𝑁 are F U N distinct points on the same line and 𝐹𝑈 + 𝑈𝑁 = 𝐹𝑁 and 2cm + 3cm = 5cm 2𝑐𝑚 + 3𝑐𝑚 = 5𝑐𝑚. 3. Definition of Collinear Points and Coplanar Points When points are on the same line, they are called collinear points. D A B C 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. 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. In the figure, points J, O, Y and line 𝑛 are all on the same plane 𝑆, hence they are coplanar. 12 CO_Q3_Mathematics 8_ Module 2 4. Definition of a Ray Ray is a part of a line that has one endpoint and goes on infinitely in one direction. In the figure above, ray PQ starts from point P and goes on to the right without bound. P is called the endpoint of 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. The common endpoint being shared by 𝑈𝑇 and 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. 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°. 13 CO_Q3_Mathematics 8_ Module 2 8. Definition of Adjacent Angles Adjacent angles share a common vertex and a common side, but do not overlap. ∠ABD and ∠CBD are ∠EFG and ∠GFH have F as adjacent angles which have a common vertex, and 𝐹𝐺 as common common vertex B and a common side, but the interiors of the two side 𝐵𝐷. The interiors of ∠ABD angles intersect, this means that and ∠CBD do not intersect and the two angles have common therefore, the two angles have no interior points. Thus, ∠EFG and interior points in common. ∠GFH are not adjacent angles. 9. Definition of Supplementary Angles Two angles are supplementary when the sum of their angles is 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. 𝐵𝐷 is the common side, 𝐵𝐴 and 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. ∠1 and ∠3 are vertical angles. ∠2 and ∠4 are also vertical angles. Noticed that these angles are opposite each other. 14 CO_Q3_Mathematics 8_ Module 2 12. Definition of Perpendicular Lines Perpendicular lines are two lines that intersect to form a right angle. ⃡ 𝐴𝐵 intersects 𝐶𝐷⃡ at point E. A ∠𝐴𝐸𝐷, ∠𝐴𝐸𝐶, ∠𝐵𝐸𝐶, and ∠𝐵𝐸𝐷 are right angles form by these E ⃡ two intersecting lines, hence, 𝐴𝐵 C D ⃡ is perpendicular to 𝐶𝐷. In ⃡ ⊥ 𝐶𝐷 symbol, 𝐴𝐵 ⃡. B 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. P ̅̅̅̅ ⊥ 𝑋𝑍 𝑃𝑅 ̅̅̅̅ at point 𝑌. 𝑃𝑅 ̅̅̅̅ divides 𝑋𝑍 ̅̅̅̅ into two equal parts. Thus, 𝑋𝑌 = 𝑍𝑌. Furthermore, 𝑋𝑌 ̅̅̅̅ ≅ 𝑍𝑌 ̅̅̅̅. X Y Z R 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. B D A C E Polygon Not Polygon 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. B F If each diagonal, except G C its endpoints, is entirely Exterior in the interior of the Interior A E polygon, then the D H polygon is convex, like polygon EFGH. 15 CO_Q3_Mathematics 8_ Module 2 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. K ⃡ which contains ̅̅̅̅ 𝐿𝑀 𝐿𝑀 also J L contains points in the interior of the polygon, hence, polygon JKLM M is nonconvex. 17. Definition of Regular Polygon A regular polygon is a polygon that is both equilateral and equiangular. E Exterior Interior E angle angle R O A R A Central O angle D D M M N The regular polygon DREAM has five Regular polygon DREAM interior angles ∠𝑅𝐷𝑀, ∠𝐸𝑅𝐷, ∠𝐴𝐸𝑅, ∠𝐸𝐴𝑀, ∠𝐴𝑀𝐷. has also five exterior angles. These angles are equal, These angles are obtained when one of the intersecting sides is 𝑚∠𝑅𝐷𝑀 = 𝑚∠𝐸𝑅𝐷 = 𝑚∠𝐴𝐸𝑅 = 𝑚∠𝐸𝐴𝑀 = 𝑚∠𝐴𝑀𝐷. extended such as ∠𝐴𝑀𝑁. The outside angle along with the The polygon DREAM has five equal sides, vertex is an exterior angle. 𝐷𝑅 = 𝑅𝐸 = 𝐸𝐴 = 𝐴𝑀 = 𝑀𝐷. Point O is the center of the given polygon. ∠𝑂 is the central angle. 18. Definition of a Triangle A triangle is a three-sided polygon. R 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 T I ̅̅̅̅, ̅̅̅̅ sides (𝑇𝑅 𝑇𝐼), three angles (∠𝑇, ∠𝑅, ∠𝐼), 𝑅𝐼, ̅̅̅̅ and three vertices (T, R, I) 16 CO_Q3_Mathematics 8_ Module 2 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. O ̅̅̅̅ is an angle bisector of ∆𝐿𝑂𝑉. It 𝑂𝐸 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 L V opposite side. Thus, ∠𝐿𝑂𝐸 ≅ ∠𝐸𝑂𝑉. E 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. I T T S S S Y A E F A L N E Every triangle has ̅̅̅̅ is ̅̅̅̅, In ∆𝑆𝐴𝐹, 𝑆𝐸 In ∆𝐿𝑆𝐸, 𝐿𝐼 , 𝑆𝑁 three altitudes. In ̅̅̅̅ are the one of the three and 𝑇𝐸 ∆𝑆𝑇𝐴, ̅̅̅̅ 𝑇𝑌 is one of altitudes. three altitudes. 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. R Every triangle has also three medians. Median, except its endpoints, is always in the triangle’s O M interior. Unlike altitude that can be drawn from the exterior of the triangle. ̅̅̅̅ 𝐴𝑂 is the median to ̅̅̅̅, ̅̅̅̅ 𝑁𝑅 𝑅𝐿 is the median to 𝑁𝐴 ̅̅̅̅, and ̅̅̅̅̅ 𝑁𝑀 is the ̅̅̅̅. median to 𝑅𝐴 N L A 17 CO_Q3_Mathematics 8_ Module 2  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. O D S D H T R Y U N Y A Acute Right Obtuse Equiangular Triangle Triangle Triangle Triangle 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. E B A H R G I H T Scalene Isosceles Equilateral Triangle Triangle Triangle ̅̅̅̅ ≅ 𝐵𝐼 𝐵𝐺 ̅̅̅ ̅̅̅̅ ≅ 𝐴𝑇 𝐻𝐴 ̅̅̅̅ ≅ 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. 18 CO_Q3_Mathematics 8_ Module 2 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. Axiom 2. If equals are added to equals, the wholes are equal. This is addition property of equality. B Axiom 3. If equals are subtracted from equals, the remainders are equal. This is subtraction property of equality. Axiom 4. Things which coincide with one another are equal to one another. This is reflexive property. - - 19 CO_Q3_Mathematics 8_ Module 2 Axiom 5. The whole is greater than the part. > 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.  Substitution Property of Equality  For all real numbers 𝑝 and 𝑞, if 𝑝 = 𝑞, then 𝑞 can be substituted for 𝑝 in any expression. ≅ Geometrical Postulates: Postulate 1. A straight line segment can be drawn joining any two distinct points. Postulate 2. Any straight line segment can be extended indefinitely in a straight line. Postulate 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 20 CO_Q3_Mathematics 8_ Module 2 Postulate 4. All right angles are congruent. Postulate 5. If a straight line meets two other lines, so as to make the two interior angles on one side of it together less than two right angles, the other straight lines will meet if produced on that side which the angles are less than two right angles. Postulate 5 asserts that two distinct straight lines in a plane are either parallel or meet exactly in one point. Postulate 5 is referred to as the parallel line postulate. This postulate was questions by many mathematicians. To them, the postulate seemed less obvious that the others and claimed that it should be proven rather than simply accepting it as a fact. The following are other postulates about point, line, plane, and figures formed by these and the basic postulates.  If two distinct planes intersect, then their intersection is a line. 21 CO_Q3_Mathematics 8_ Module 2   If two points of a line are in a plane, then the line containing these points is in the plane.  Segment Addition Postulate  If points 𝑃, 𝑄 and 𝑅 are collinear and point 𝑄 is between points 𝑃 and 𝑅, then 𝑃𝑄 + 𝑄𝑅 = 𝑃𝑅  Angle Measurement Postulate  To every angle there corresponds a unique real number 𝑟 where 0 < 𝑟 < 180°. In geometry, angles are measures in units called degrees. In symbol °. Angle FUN is 45 degrees. In notation, 𝑚∠𝐹𝑈𝑁 = 45°  Angle Addition Postulate ° ° ° A System of Logic (Proof) The sets of axioms (or postulates) you just learned were used to deduce new propositions or to prove other statements using the rules of inference of a system of logic. In other words, the system of logic is your proof. Euclid used deductive reasoning in organizing the Euclidian geometry. 22 CO_Q3_Mathematics 8_ Module 2 Theorems Theorems are the new statements which are deduced or proved using sets of axioms, system of logic, and previous theorems. These are statements accepted after proven deductively. The following are some theorems about points, lines, and planes. Theorems Descriptions Illustrative Example If two different lines intersect, Theorem 1 then they intersect at exactly one point. If a line not contained in a plane intersects the plane, then the Theorem 2 intersection contains only one point. If a point lies outside a line, then Theorem 3 exactly one plane contains both the line and the point. If two distinct (different) lines Theorem 4 intersect, then exactly one plane contains both lines. Linear Pair If two angles form a linear pair, Theorem then they are supplementary. ° Vertical Angles Vertical angles are congruent. Theorem 23 CO_Q3_Mathematics 8_ Module 2  Point on the perpendicular Perpendicular bisector of a segment is Bisector Theorem equidistant from the endpoints of the segment. Isosceles Triangle Base angles of isosceles triangles Theorem are congruent. The exterior angle of a triangle is equal to the sum of the two remote interior angles of the Exterior Angle triangle. The exterior angle of a Theorem triangle is greater than either of the measures of the remote interior angles. Triangle Angle The sum of the measures of the Sum Theorem angles of a triangle is 180°. Now that you have learned definitions, postulates and theorem, you are now ready for the next activities. 24 CO_Q3_Mathematics 8_ Module 2  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 o Exterior Angle 𝑚∠𝑡 = 𝑚∠ℎ + 𝑚∠𝑜 Theorem h t 1. 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶 A B C 2. R 𝑚 ∠ 𝑅𝐴𝐶 + 𝐶𝐴𝐸 C = A E 𝑚 ∠ 𝑅𝐴𝐸 3. O 𝑚 ∠ 𝑃𝑁𝑂 + 𝑚 ∠ 𝑂𝑁𝑌 P N Y 25 CO_Q3_Mathematics 8_ Module 2  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 Statement Reason Figure M C 1. 𝑚∠𝑀𝐴𝑃 = 𝑚∠𝐶𝐴𝑅 1. 2. 𝑚∠𝑀𝐴𝑃 = 𝑚∠1 + 𝑚∠2 2. P 1 3. 𝑚∠𝐶𝐴𝑅 = 𝑚∠2 + 𝑚∠3 3. 2 3 4. 𝑚∠1 + 𝑚∠2 = 𝑚∠ 2 + 𝑚∠3 4. A 5. 𝑚∠2 = 𝑚∠ 2 5. R 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 Illustration Postulate 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. 26 CO_Q3_Mathematics 8_ Module 2  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. 27 CO_Q3_Mathematics 8_ Module 2  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 C. plane B. line D. angle 2. What undefined term is represented by the edge of a ruler? A. point C. plane B. line D. angle 3. Which of the following does not represent a plane? A. tip of a pen C. cover page of a book B. top of the table D. faces of a rectangular box 4. A consistent axiomatic system does NOT do what to itself? A. doubt C. affirm B. guess D. contradict 28 CO_Q3_Mathematics 8_ Module 2 5. Which of the following statements best describe a perpendicular line? A. a straight line C. intersecting lines forming a linear pair B. intersecting lines 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 C. congruent segments B. line segment D. perpendicular bisector 7. What axioms of equality do the illustration below represent? Given: EM ≅ EM A. reflexive property C. symmetric property B. substitution property D. transitive property 8. Which of the following represents a plane? A. cover page of ADM module C. corner of a room B. edge of the of the book 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 C. symmetric property B. substitution property D. transitive property 10. What is the sum of all interior angles of a triangle? A. 90° C. 180° B. 160° D. 360° 11. Which of the statements justify a supplementary angle? A. It has a common side. C. The sum of two angles is 180°. B. It has the same measures. 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 C. building and campus B. sidewalk only D. building and sidewalk 29 CO_Q3_Mathematics 8_ Module 2 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 b 3 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 C. III, II, I, IV B. II, IV, I, III 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 C. isosceles and equilateral triangles B. scalene and isosceles triangles D. The materials cannot form a triangle 30 CO_Q3_Mathematics 8_ Module 2  Additional Activities Directions: Complete the table below with the facts associated with the given problems. Facts Application/ Illustration Solution Vertical angles Use the facts that linear pair 1. are congruent. forms supplementary angles to 2. prove that vertical angles are congruent. 3. 4. 5. x w 6. y l z 7. m 8. 9. 31 CO_Q3_Mathematics 8_ Module 2 CO_Q3_Mathematics 8_ Module 2 32 What's More What I Have Learned What I know Activity 1: Make me meaningful! 1. T 1. A 9. B 1. line and point 2. F – using only 2. C 10. D 2. a. axioms, a logical 3. D 11. D Axioms Model 1 Model 2 system, and previous 4. D 12. A 1   theorems 5. D 13. C 2   3. T 6. B 14. A 3   4. F – cannot be used 7. B 15. C 5. T 8. B b. The two models are non-isomorphic because 6. F – are not necessary they are not of equivalent structure or they 7. T What’s In do not look exactly the same. 8. T 1. White color c. The axiomatic system is consistent because 9. F – common notions (Commutative) the two models are non-isomorphic making are none geometric, 2. Violet (Existence of the system non-categorical, hence, it is a postulates are Multiplicative consistent system. geometric Identity) 10. T 3. Red (Trichotomy Activity 2 Axiom) What I can Do 4. Yellow (Associative 1. Segment Addition Postulate Axiom) 2. Angle Addition Postulate 5. Blue (Transitive 1. committee and Axiom) 3. Definition of Supplementary angles member 2. Possible model What’s New Activity 3 Points 1. Given  Corner of a 2. Angle addition Postulate rectangular tray 3. Angle addition Postulate  Tip of a marker 4. Substitution property 3. By axiom 1, a  Tip of a pencil 5. Reflexive property member exist. Lines By axiom 2, there  Stretched rope Activity 4  Edge of the are exactly 6 cabinet committees.  Edge of a ruler Therefore, there  Edge of a book is a finite number Plane of members.  Top of the table  Wall of a Assessment classroom  Cellphone screen 1. A 2. B 3. A 4. D 5. D 6. D Additional Activities 7. A 8. C 1. m∠w + m∠x = 180° 9. C 2. m∠y + m∠z = 180° 10. C 11. C 3. m∠w + m∠x = m∠y + m∠z 12. D 4. m∠w = m∠y 13. B 5. m∠x + m∠y = 180° 14. A 6. m∠z + m∠y = 180° 15. C 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 Answer Key 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.ht m 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-triangle- inequality-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 33 CO_Q3_Mathematics 8_ Module 2 For inquiries or feedback, please write or call: Department of Education - Bureau of Learning Resources (DepEd-BLR) Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600 Telefax: (632) 8634-1072; 8634-1054; 8631-4985 Email Address: [email protected] * [email protected]