Mathematics Grade 8 Part 2

MATHEMATICS Teacher's Guide Grade 8 Part 2



TEACHING GUIDEModule 6: Reasons Behind ReasoningA. Learning Outcomes Content Standard: The learner demonstrates understanding of key concepts of axiomatic development of geometry. Performance Standard: The learner is able to communicate mathematical thinking with coherence and clarity in formulating, investigating, analyzing and solving real life problems.UNPACKING THE STANDARDS FOR UNDERSTANDINGSUBJECT: LEARNING COMPETENCIESGrade 8 Mathematics 1. Identify the hypothesis and conclusions of if-then and other types of statements.QUARTER: 2. Formulate the inverse, converse, and contrapositive of an implication.Third Quarter 3. Distinguish between inductive and deductive reasoning. 4. Provide formal arguments that explain results of phenomenon or a situation.STRAND: 5. Write formal arguments as a series of statements that make up a proof (both directGeometry and indirect).TOPIC: 6. Explain the need for defined terms previously introduced.Reasoning and Proofs 7. Differentiate between postulate and theorem.LESSONS: ESSENTIAL UNDERSTANDING: ESSENTIAL QUESTION:1.If-then Statement Logic and reasoning are tools in geometry to facilitate How do you establish valid2.Inductive and Deductive Reasoning mathematical thinking for making valid conclusions. conclusions?3.Writing Proofs The use of inductive and/or deductive reasoning depends When do we use inductive on the given situation. and/or deductive reasoning?TRANSFER GOAL:Students will on their own make valid conclusion and communicate mathematical thinking with coherence and clarity on a real-lifesituation. 341

B. Planning for AssessmentProduct/PerformanceThe following are products and performances that students are expected to accomplish with in this module. Assessment Map KNOWLEDGE UNDERSTANDING TRANSFER AND PROCESS/ (MEANING MAKING) TYPE SKILLS Pre–test Pre–test PRE-ASSESSMENT/ (ACQUISITION) Inbox - Outbox Sheet DIAGNOSTIC Pre–test Interpretation, Explanation Written Exercises Interpretation, Explanation, Picture Translation Sheet Perspective Written Exercises Case Solved! Activity Sheet Interpretation, Explanation Quiz Interpretation, Explanation, Perspective Investigating Cases Interpretation, ExplanationFORMATIVE ASSESSMENT Prove it! Interpretation, Explanation REAL Proving Interpretation, Explanation One – minute essay Explanation, Self – knowledge Self – knowledge Rubric on the Article in newspaper 342

SUMMATIVE Post Test Inbox – Outbox Sheet Rubric on the Performance ASSESSMENT Interpretation, Explanation taskSELF-ASSESSMENT Post Test Post Test Synthesis Journal Assessment Matrix (Summative Test) Levels of What will I assess? How will I assess? How Will I Score?Assessment Knowledge The learner demonstrates Paper and Pencil Test 1 point for every correct response 15% understanding of key concepts of 1 point for every correct responseProcess/Skills axiomatic development of geometry. 25% Identify the hypothesis and conclusions Paper and Pencil Test RubricUnderstanding of if-then and other types of statements. Criteria: 30% Relevance Formulate the inverse, converse, and Creativity Insightful contrapositive of an implication. Distinguishes between inductive and Journal writting deductive reasoning. Provides formal arguments that explain results of a phenomenon or a situation Portfolio Write formal arguments as a series of statements that make up a proof (both direct and indirect). The learner is able to formulate/ solve GRASPS Assessment Rubric on Mathematical real-life problems involving reasoning. Make a mathematical Investigation investigation that willProduct enlighten the readers by Criteria: 30% providing valid conclusions. The written output of the investigation shall be evaluated Apply your understanding based on to its coherence, clarity, of the key concepts of judgment, and mathematical reasoning. reasoning. 343

C. Planning for Teaching-LearningIntroduction: The learner demonstrates understanding of the key concepts of axiomatic development of geometry. It consists of threelessons namely: Lesson 1 – If- then Statements Lesson 2 – Inductive and Deductive Reasoning Lesson 3 – Writing Proof As an introduction to the main lesson, ask them the following questions: Is it possible to make a valid conclusion withouteven going through the process of investigation? What will you do if you were asked to make a decision that will affect manypeople? Many aspects in our life involve decisions and proof. In all lessons, students are given the opportunity to use their prior knowledge and skills in learning proofs in geometry. They are also given varied activities to process the knowledge and skills learned and to deepen and transfer theirunderstanding of the different lessons. Have you ever asked yourself how do you establish valid conclusions? When do we use inductive and/or deductive reasoning? Entice the students to find out the answers to these questions and to determine the vast applications of proofs throughthis module.Objectives: After the learners have gone through the lessons contained in this module, they are expected to:  Identify the hypothesis and conclusions of if-then and other types of statements.  Formulate the inverse, converse and contrapositive of an implication.  Distinguish between inductive and deductive reasoning.  Provide formal arguments that explain results of a phenomenon or a situation.  Use syllogism in writing formal arguments as a series of statements that make up a proof.  Explain the need and importance for defined terms previously learned.  Differentiate between postulate and theorem. 344

Pre-Assessment: Choose the letter of the correct answer1. Which of the following best describes deductive reasoning? a. using logic to draw conclusions based on accepted statements b. accepting the meaning of a term without definition c. defining mathematical terms in relation with physical objects d. inferring a general truth by examining a number of specific examples Answer: A 2. Theorem: A triangle has at most one obtuse angle. Francisco is proving the theorem above by contradiction. He began by assuming that in ∆ABC, ∠A and ∠B are both obtuse. Which theorem will Francisco use to reach a contradiction? a. If two angles of a triangle are congruent, the sides opposite the angles are congruent. b. If two supplementary angles are congruent, each angle measures 90°. c. The largest angle in a triangle is opposite the longest side. d. The sum of the measures of the angles of a triangle is 180°. Answer: D3. If m∠R + m∠M = 90° then a. ∠R ≅ ∠M. b. ∠R and ∠M are right angles. c. ∠R and ∠M are complementary. d. ∠R and ∠M are supplementary. Answer: C4. The converse of the statement:” if you are in love then you are inspired”, is, a. If you are not in love, then you are not inspired. b. If you are inspired, then you are in love. c. If you are not inspired, then you are not in love. d. if you are in love, you are not inspired. Answer: B 345

5. The if-then form of the statement: \"Parallel lines never intersect”, is: a. If two lines intersect, then they are parallel. b. If two lines are parallel, then they never intersect. c. if two lines are not parallel then they intersect. d. If two lines intersect, then they are not parallel. Answer: B 6. What is the inverse of the statement,:” If the number is divisible by 2 and 3, then it is divisible by 6”. a. If the number is divisible by 6, then it is divisible by 2 and 3. b. If the number is not divisible by 2 and 3, then it is not divisible by 6. c. If the number is not divisible by 6, then it is not divisible by 2 and 3. d. If the number is divisible by 2 and 3, then it is not divisible by 6. Answer: B 7. What property is illustrated in : If ∠A ≅ ∠B, ∠B ≅ ∠C then ∠A ≅ ∠C. a. Reflexive Property b. Symmetric Property c. Transitive Property d. Addition Property Answer: C 8. Using the distributive property, 4 (a + b) = _________. a. 4a + b b. B + 4a c. 4a + 4b d. 4 + a + b Answer: C 9. Supply a valid conclusion for the given hypothesis: if O→M bisects ∠LON then a. ∠LOM ≅ ∠NOM b. ∠LOM ≅ ∠LON 346

c ∠MON ≅ ∠NOL d. m∠LON = m∠LOM + m∠MON Answer: A10. The method of proof by contradiction is: a. direct proof b. formal proof c. indirect proof d. two column proof Answer: c11. If garbage are disposed properly then dengue diseases will be prevented. What is the underlined portion called in the conditional statement? a. the conclusion b. the hypothesis c. the argument d. the converse Answer: B 12. How many dots are there in the three figure? Figure 1 Figure 2 Figure 3 Answer: 20 347

13. Which of the following statements is true? a. If ∠1 has a measure of 90∠1,then ∠1 is obtuse. b. If ∠1 has a measure of 140°, then ∠1 is acute. c. If ∠1 has a measure of 35°, then ∠1 is acute. d. If ∠1 has a measure of 180°,then ∠1 is right. Answer: C 14. Which of the following statements is false? a. Any four non-collinear points lie in a distinct plane. b. A plane contains at least 3 non-collinear points. c. Any two lines intersect at a point. d. Through two given points we can draw three lines. Answer: D 15. Rewrite the statement in if-then form. a. A figure has four sides if and only if it is a quadrilateral. b. If a figure is a quadrilateral, then it has four sides. c. If a figure has four sides, then it is a quadrilateral. d. A figure is a quadrilateral if and only if it has four sides. Answer: B 16 Name the property which justifies the following conclusion. Given : JB = 28 Conclusion : JB + 4 = 32 a. Addition property of equality b. Multiplication property of equality c. Substitution property of equality d. Transitive property of equality Answer: A 348

For 17-20 Give the answer. 17. ∠1 and ∠2 are complementary angles. ∠1 and ∠3 are vertical angles If m∠3 = 49°, find m∠2. Answer: m∠2 = 4 18. What is the missing reason in the following proof? m∠1 = m∠3, m∠2 = m∠3 Answer: Transitive Property 1 2 3 Statement Reason 1. m∠1 = m∠3, m∠2 = m∠3 1. given 2. m∠2 = m∠3 2. ___?____ 19. Supply the missing statement in the following proof. Given: m∠1 + m∠2 = m∠2 + m∠3 Answer: m∠2 = m∠2 1 2 Prove; m∠1 = m∠3 3 Statement Reason 1. m∠1 + m∠2 = m∠2 + m∠3 1. given 2. Reflexive Property 2. ----------?----------- 3. Subtraction property 3. m∠1 = m∠3 20. What conclusion can be logically deduced based on the following statements? If you are catholic, you are against the RH Bill Mrs. Romano is a Catholic. Answer: Mrs. Romano is against RH Bill 349

WWhhaatt ttoo KKnnooww 1Lesson If- then Statements Ask the students to accomplish the activity sheet called INBOX – OUTBOX sheet.Teacher’s Note and Reminders WWhhaatt ttoo KKnnooww Let’s begin this lesson by accomplishing the activity sheet below called INBOX – OUTBOX sheet. Activity 1 INBOX – OUTBOX SHEET Description: This activity is intended to elicit your prior knowledge regarding the Direction: lesson. Answer the question below and write your answer in the space provided IN THE BOX. How do you make valid conclusions if faced with problems in life such as having failing grades, meeting deadlines and even in love- life troubles? Don’tForget! 350

The students gave their initial ideas on how to make sound judgment and Activity 2 ARTICLE ANALYSIShow useful it was. Now let them find out the other answers by doing the nextpart. What they will learn in the next sections will also enable them to do the Description: There are things in life which involve decision-making. Find out how validfinal project which involves investigating mathematical concepts. Direction: decision making affects our life. The given article below deals with the effect of having or giving misguided conclusion. Teacher’s Note and Reminders Read the excerpts on the article from Bombo Radio Philippines entitled, “Judge sinibak ng SC due to wrong decisions” then answer the follow – up questions below. Judge sinibak ng SC due to wrong decisions Revoked of license by the Supreme Court (SC) is a judge in Cotabato City because of the issuance of a decision in the case of annulment of marriage without conducting a hearing. In the per curiam decision of the Supreme Court en banc discharged on May 15, 2012, proven guilty of gross misconduct and dishonesty is Judge Cader Indar Al Haj, presiding judge of the Regional Trial Court (RTC), Branch 14, Cotabato City and also serve as acting presiding judge of RTC, Branch 15, Shariff Aguak, Maguindanao. Along with its disbarment Indar was removed from the service and also the benefits that he should get from his retirement except leave credits. Also stated in the decision was Indar can’t work in any office of the government owned including any government owned and controlled corporations. Don’t According to the court, Indar violated the Canons 1 and 7, as well paki check poForget! as Rule 1.01 of the Code of Professional Responsibility, thus, also ito ung sa LM removing his name on the roll of attorneys. unti lang sa TG madami The penalty of Indar stemmed from a report sent by the LCR of Manila and Quezon City to the Office of Court Administration (OCA) in relation to the decisions resolutions and orders in the marriage annulment issued by the same judge. http://www.bomboradyo.com/story-of-the-day/111056-judge-sinibak- ng-sc-dahil-sa-maling-desisyon This site provided the article above entitled ”Judge sinibak ng SC due to wrong decisions” 351

WWhhaatt ttoo PPrroocceessss QU ?E S T I ONS 1. Comment on the article. 2. Site a situation where important decision making is needed. The goal in this section is to learn and understand key concepts of 3. Suggest a procedure on how to make a wise decision. reasoning and proving. The students will be dealing with If-then statement, deductive and inductive reasoning and writing proofs. You gave your initial ideas on how to make sound judgment and how useful Let them do Activity 3. it was. Let’s now find out the other answers by doing the next part. What have you learned. In the next sections will also enable you to do the final project which involves Allow students to write something about the statements in their journal. investigating mathematical concepts. Tell them that they have just encountered conditional statements or the If- then statements. Let them read some notes about an if-then statement. WWhhaatt ttoo PPrroocceessss Teacher’s Note and Reminders Your goal in this section is to learn and understand key concepts of reasoning and proving. You will be dealing with If-Then statement, Deductive and Inductive reasoning and writing proofs. Activity 3 JUDGE US! Description: A lot in the statements that we encounter are logically constructed or Direction: written but NOT valid or acceptable. This activity deals with determining which statement is valid or not. From the given statements tell whether the statement is valid or not. Don’t 1. Students who are good in mathematics are smart.Forget! Enchong is smart, then he is good in mathematics. 2. Young actresses are health conscious. Kim is a young actress then she is health conscious. 3. If it rains then the sports fest will be cancelled. It rains therefore the sports fest is cancelled. 4. If the lines are parallel they do not intersect Line x and line y do not intersect; therefore they, are parallel. 5. If two angles are right angles ,then they are congruent. ∠A and ∠B are congruent, then they are right angles. For items 6 to 10 complete the statement and justify your answer 6 . Miss Earth candidates are environmentalists. Miss Jaybee is a candidate to the Miss Earth search, therefore 7. If you are at SM you got it all. Marie is at SM then 352

Teacher’s Note and Reminders QU NS 8. If you bank with BDO they find ways. Vincent has deposit at BDO then Don’t ?E S T I O Forget! 9. If you drink coke you find happiness. Jay is drinking coke then… 10. Globe connects people. Dedeth is using globe simcard then.. a. What have you noticed about the statements given above? b. Take one of the statements and tell something about it c. .What is common to all of the statements? Write your answers in your journal and have a small discussion with your group. You have just encountered conditional statements or the If-then statements. An if-then statement is composed of two clauses: the if- clause and the then- clause. We can denote a letter for each clause, p for the if clause and q for the then clause. The statement is in the form” If p then q. Conditional statements are formed by joining two statements p and q using the words if and then. The p statement is called the hypothesis and the q statement is the conclusion. A simple flow of reasoning from if-clause to the then-clause is called simple implication. There are some conditional statements not written in this form but you can rewrite them using the if-then form. How will you identify the hypothesis and the conclusion? You try this: 1. Cigarette smoking is dangerous to your health. If-then form ________________________________________ Hypothesis ________________________________________ Conclusion ________________________________________ 2. It is more fun in the Philippines. If-then form ________________________________________ Hypothesis ________________________________________ Conclusion ________________________________________ 3. A segment has exactly one midpoint. If-then form ________________________________________ Hypothesis ________________________________________ Conclusion ________________________________________ 353

Answer Key 4. Angles in a linear pair are supplementary.Exercise 1 If-then form ________________________________________1. If the polygon is a rectangle then the opposite sides are parallel Hypothesis ________________________________________2. If you are Filipino then you are God-fearing people. Conclusion ________________________________________3. If two angles are complementary then the sum is 90°.4. If Paolo is a good citizen then he obeys rules and regulations. 5. Vertical angles are congruent.5. If the polygon is a triangle then it has three sides. If-then form ________________________________________ Hypothesis ________________________________________ Teacher’s Note and Reminders Conclusion ________________________________________ Don’t How do you distinguish the hypothesis from the conclusion when the statement is Forget! not in the if-then form? See the examples below. 1. National Disaster Risk Reduction Council volunteers are busy during calamities. 2. An eighteen year old Filipino can cast his/her vote during election. 3. All right angles are congruent. 4. Three non-collinear points determine a plane. 5. Perpendicular lines are intersecting lines. Discuss with a partner the underlined part of the sentence and the one in bold letters. What part of the sentence are the underlined words ? What part of the sentence are in bold letters? Which is the hypothesis and which is the conclusion? Rewrite the statements to if-then form. Now that you know what conditional statements are, and you can identify the hypothesis and the conclusion, have more practice in answering the exercises below. Exercise 1 Convert each statement in if-then form, then identify the hypothesis and the conclusion. 1. Opposite sides of a rectangle are parallel. 2. Filipinos are God-fearing people. 3. The sum of the measures of complementary angles is 90°. 4. Good citizens obey rules and regulations. 5. A triangle is a polygon of three sides. 6. A quadrilateral has four sides. 7. Two points determine a line. 8. The intersection of two lines is a point. 9. Two intersecting lines line in one plane. 10. The sum of the angles forming a linear pair is 180° Now that you are well- versed in converting conditional statement to if-then form, you can easily identify the hypothesis and the conclusion. When do you say that the implication is true or false? 354

Teacher’s Note and Reminders The implication p → q is always true except in the case that p is true and q is false. See the truth table for implication below. Don’t Forget! p q p→q TTT TFF FTT FFT This time let us make another statement from the given one. Let us do the activity. Activity 4 JUMBLED WORDS Direction: Make a sentence from the jumbled words. POLYGON TRIANGLE A IS A 1. A TRIANGLE IS A POLYGON. 2. A POLYGON IS A TRIANGLE. 355 Let’s take the #1 sentence as our first statement. First we can convert it to if-then form, then we can form its converse, inverse, and contrapositive. Study the table below . Statement If-then form Converse Inverse Contrapositive • A triangle is If an object is a If an object is a If an object is If an object is a polygon. triangle, then it polygon , then it not a triangle’ not a polygon, is a polygon is a triangle. then it is not a then it is not a polygon. triangle. Discuss with your group how the converse is written? Inverse? and contrapositive of a given statement. If p is ; If the object is a triangle q is : then it is a polygon What happen to p and q in the converse? Compare the inverse and the original statement. What did you do with p? what did you do with q?

Answer Key Observe the changes in the contrapositive.Exercise 2 Summarize your observation in terms of p and q.A. Converse: If the angles have the same measure then they are congruent. Let’s take another statement: An even number is divisible by two. Inverse: If two angles are not congruent then they do not have the same measure If-then form ________________________________________________ Contra positive: If two angles do not have the same measure then they Converse ________________________________________________ are not congruent. Inverse ________________________________________________ Contrapositive ________________________________________________B. 1. If three points are not collinear, then they determine a plane. 2. If an object is a rectangle, then it has four right angles. We can summarize how to convert the statement in terms of p and q. 3. If two lines are perpendicular, then they intersect each other. See the table below. Teacher’s Note and Reminders Statement If p, then q Converse If q, then p Inverse If not p, then not q Contrapositive If not q, then not p Exercise 2 A. Fill up the table below. If two angles are congruent, then they have the same measure. Statement Converse Inverse Contrapositive. B. State the converse of the following statements: 1. Three non-collinear points determine a plane. 2. A rectangle has four right angles. 3. Perpendicular lines intersect. Don’t Go back to Activity 4 .Forget! If p, then q : If an object is a triangle then it is a polygon. Converse If q, then p: If an object is a polygon then it is a triangle. Analyze the converse. Is it true? If not, give a counter example. The converse is false because square is a a polygon. It is not a triangle. 356

Now that the students can identify the hypothesis and the conclusion in the If p, then q: If a number is even then it is divisible by two.if-then statement, and form its converse, inverse, and contrapositive, they If q, then p: If a number is divisible by two then it is even.are now ready to study the kinds of reasoning in the next section. The converse is true. Teacher’s Note and Reminders Try to analyze the converse of the statements in B. So what can you conclude about the converse of a statement? Is the converse of a given statement always true? Activity 5 “PICTURE ME” Don’t Observe the set of pictures. Translate the pictures into conditional statements. StateForget! the converse, inverse and contrapositive of the conditional statements. Classify each as true or false and justify. Go to other group, share each other's answers and come up with a common conclusion. Now that you can identify the hypothesis and the conclusion in the if-then statement, and form its converse, you are now ready to study the kinds of reasoning in the next section . 357

Let them discuss with their group their answers to these questions. 2Lesson Inductive and• How did you arrive at your answer? Deductive Reasoning• Did you agree at once on your answer?• Were there disagreements among the members? Activity 6 WHY OH WHY?• What you have shown is inductive reasoning . Can you give three examples?• Based on the activity, what is inductive reasoning? Each group will be given this activity sheet to accomplish.The teacher can ask the students to give more examples of inductive 1. Look carefully at the figures , what is next?reasoning using the following situations:• Classroom situation 2. Study the pattern and draw the next figure.• Supermarket situation• Situation during family reunion 3. My Math teacher is strict.• JS Prom My previous math teacher was strict.• Election What can you say about all math teachers?• Summer vacation 4. 1 × 10 = 10This time tell the students that there is another kind of reasoning. Let them 2 × 10 = 20try to accomplish Activity 7 for them to discover what it is. 3 × 10 = 30 5 × 10 = 50 Teacher’s Note and Reminders 24 × 10 = 240 2345 × 10 = ______. Don’t Forget! 5. Every time Jackie visits her doctor she receives excellent services .With this she believes that..____________________________________ Discuss the following with your group • How did you arrive at your answer? • Did you agree at once on your answer? • Were there arguments among the members? • What you have shown is inductive reasoning. Give 3 examples. • Based on the activity, define inductive reasoning? Inductive reasoning takes specific examples to make a general rule. 358

The students have just encountered deductive reasoning. Activity 7 CUBRA CUBEAsk them to give the difference between inductive and deductive reasoning,then let them answer exercise 3. Complete the table below. ConclusionOne of the tools used in proving is reasoning, specifically deductive Statementreasoning. Deductive reasoning is a type of logical reasoning that uses 1. Filipinos are hospitable.accepted facts to reason in a step-by-step manner until we arrive at thedesired statement. Bonifacio is a Filipino.A proof is a logical argument in which each statement you make is supported/ 2. If points are collinear, then they liejustified by given information, definitions, axioms, postulates, theorems, and on the same planepreviously proven statements. Points R,M,andN are collinear .Remember:• Postulate is a statement that is accepted without proof. 3. A quadrilateral is a polygon of four• Theorem is a statement accepted after it is proved deductively. sidesAnswer Key 4. Smoking can cause cancer. Tomas isExercise 3 smoking1. 25. Inductive reasoning2. X, Y, Z are on the same plane. Deductive reasoning 5. An angle is acute if its measure is3. BELEN is equilateral. Deductive reasoning between 00 and 900. Angle B is acute.4. All teachers are ladies. Inductive reasoning5. Julia is a peace-loving person. Deductive reasoning You have just encountered Deductive reasoning. Can you give the difference between inductive and deductive reasoning? Teacher’s Note and Reminders Deductive reasoning is reasoning which begins using basic and general statements Don’t to prove more complicated statements. Forget! Inductive reasoning is judging by experience while deductive reasoning is judging by logical progression. Exercise 3 Draw conclusion from each given situation and identify the kind of reasoning used. 1. 5, 10,15, 20. The next number is ___. 2. Coplanar points are points on the same plane. X,Y, Z are coplanar. Therefore__________________ 3. Regular polygon is equilateral. BELEN is a regular pentagon. Therefore___________________ 4. A child’s teacher in pre school was a female, in his grades 1 and 2 his teachers were both female. The child may say___________ 5. Filipinos are peace- loving people. Julia is a Filipino. Therefore______- The main focus in the study of geometry is to learn how to think logically. Do you still remember the If-then statement? Which one is the hypothesis? the conclusion? 359

In the next Activity the students are asked to give conclusion and reason. The parts of a deductive reasoning are:The teacher may recall definitions, axioms, postulates and proved theorems • Hypothesis – the statement which is accepted or known at the beginningwhich are tools for deductive reasoning. • Conclusion – the statement drawn from the hypothesis. Teacher’s Note and Reminders Activity 8 LET'S CONCLUDE A . Supply the conclusion for the given hypothesis 1. If ∠1 ≅ ∠2, then __________ 2. If AB = CE, then __________ 3. If ∠B and ∠E, are complementary then __________ 4. m∠3 + m∠5 = 180, then __________ 5. If ∠A and ∠X form a linear pair, then __________ B. Supply a valid conclusion for the given hypothesis on the first blank and the corresponding reason on the second blank 6. If ∠B is a right angle, Then _______________ ______________ 7. If m∠3 + m∠4 = 180 8 . ITfh→PeMn _______________ ______________ bisects ∠APO 9 . ITfh→BenP _⊥_→B__C___________ ______________ Then _______________ ______________ 10. ∆BOS is isosceles. Then _______________ ______________ From the hypothesis we derive another statement that is the conclusion Where did you base your conclusion? Have you recalled your undefined terms, definitions and postulates? They will play a very important role in our next section. Don’tForget! 360

Teacher’s Note and Reminders 3Lesson Proving Theorems Don’t In proving theorems, the properties of equality and congruence are the bases for Forget! reasoning. Properties of Equality Addition Property of Equality (APE) For all real numbers a, b, c and d, if a = b and c = d, then a + c = b + d Subtraction Property of Equality (SPE) If a = b and c = d, then a – c = b – d. Multiplication Property of Equality (MPE) If a = b, then ac = bc Division Property of Equality (DPE) If a = b then a/c = b/c Substitution Property of Equality If m∠A = 60, m∠B = 60 then m∠A = m∠B Distributive Property a(b + c) = ab + ac Properties of Congruence Reflexive Property AB ≅ AB Symmetric Property If ∠A ≅ ∠B then ∠B ≅ ∠A 361

Ask the students to answer exercise 4 Transitive PropertyIn proving theorems you have to follow these steps: Read and understand the theorem If ∠A ≅ ∠B and ∠B ≅ ∠C then ∠A ≅ ∠C Label the hypothesis as Given ,the conclusion as Prove Draw the figure and label the parts correctly. Aside from the properties of equality and congruence, you should be equipped with Write the proof which consists of the statements and reasons. the knowledge about undefined terms, definitions, and postulates in geometry. These areAnswer Key necessary to successfully support the statement of a proof.Exercise 41. Symmetric Prop. Exercise 42. Distributive Prop. Justify each statement by giving the Property of Equality or Property of Congruence3. Substitution Prop. of Equality used.4. Reflexive Prop.5. Transitive Prop. If TX = BK, then BK = TX 8(m + n) = 8m + 8n Teacher’s Note and Reminders If CT = 12 and PR + CT = 20, then PR + 12 = 20. m∠HIT = m∠HIT Don’t If ∠S ≅ ∠P, ∠B ≅ ∠S, then ∠P ≅ ∠B Forget! One of the tools used in proving is reasoning, specifically deductive reasoning. Deductive reasoning is a type of logical reasoning that uses accepted facts to reason in a step-by-step manner until we arrive at the desired statement. A proof is a logical argument in which each statement you make is supported/ justified by given information, definitions, axioms, postulates, theorems, and previously proven statements. Remember: • Postulate is a statement that is accepted without proof. • Theorem is a statement accepted after it is proved deductively. In proving theorems you have to follow these steps: • Read and understand the theorem • Label the hypothesis as given and the conclusion as Prove • Draw the figure and label the parts correctly. • Write the proof which consists of the statements and reasons. 362

Teacher’s Note and Reminders Proofs can be written in different ways Don’t 1. Paragraph Form/ Informal Proof: Forget! The paragraph or informal proof is the type of proof where you write a paragraph to explain why a conjecture for a given situation is true. Given: ∠LOE and ∠EOV L are complementary Prove: LO ⊥ OV E OV Proof: Since ∠LOE and ∠EOV are complementary, then m∠LOE + m∠EOV = 90° by definition of complementary angles. Thus, m∠LOE + m∠EOV = m∠LOV by angle addition postulate and m∠LOV = 90° by transitive property of equality. So, ∠LOV is a right angle by definition of right angles; and therefore, LO ⊥ OV by definition of perpendicularity. S 2. Two-Column Form/ Formal Proof: Given: m∠SEP = m∠TER E 1 T 2 Prove : m∠1 = m∠3 3 P Write the missing reasons R Proof: Reason 1. Statement 2. Angle Addition Postulate 1. m∠SEP = m∠TER 3. 2. m∠SEP = m∠1 + m∠2 4. Substitution Property 3. m∠TER = m∠2 + m∠3 5 4. m∠1 + m∠2 = m∠2 + m∠3 6. Subtraction Property 5 m∠2 = m∠2 6. m∠1 = m∠3 Study carefully the parts, especially the proof. How do we derive the statements and the reasons. Try the flow chart form using the same example. 363

Teacher’s Note and Reminders 3. Flowchart Form: Don’t A flowchart-proof organizes a series of statements in a logical order, Forget! starting with the given statements. Each statement together with its justification is written in a box. Arrows are used to show how each statement leads to another. It can make ones logic visible and help others follow the reasoning. Example 1 E Flow Chart Proof R C Given: RA ≅ RE CE ≅ CA Prove: ∠E ≅ ∠A 1. RA ≅ RE A 2. CE ≅ CA 4. ∆RAC ≅ ∆REC 5. ∠E ≅ ∠A 3. RC ≅ RC Example 2 CPCTC m∠SEP = m∠TER m∠SEP = m∠1 + m∠2 Angle Addition Postulate Given: m∠TER = m∠2 + m∠3 Angle Addition Postulate m∠1 + m∠2 = m∠2 + m∠3 Substitution m∠2 = m∠2 m∠1 = m∠3 Reflexive Property SPE You might want to watch a video lesson on this kind of proof, you may visit the following link: http://www.youtube.com/watch?feature=player_embedded&v=3Ti7-Ojr7Cg 364

Answer Key 4. Indirect ProofExercise 5Given: a || b with transversal t cutting a and b. An indirect proof usually is paragraph form, the opposite of the statement toFigure: be proven is assumed true until the assumption leads to contradiction. Example: E 18 a Example :27 36 b Given: ∆BEL is isosceles triangle with vertex ∠B45 Prove: ∠B ≅ ∠Lt Proof: Assume that ∠B ≅ ∠L Given that ∆BEL is isosceles therefore B L BE ≅ BL by the definition of isosceles triangle ∠B ≅ ∠L because if two sides of a triangle are congruent then theProve: ∠2 ≅ ∠6 angles opposite these sides are congruent; thus, the assumption isProof: false and therefore ∠B ≅ ∠L. Statement Reason Exercise 5 m∠2 + m∠7 = 180 Work in group m∠7 + m∠6 = 180 1. Definition of Supplementary angles m∠2 + m∠7 = m∠7 + m∠6 Prove that if two parallel lines are cut by a transversal, then the alternate interior m∠7 = m∠7 2. Transitive Prop. m∠2 = m∠6 3. Reflexive Prop. angles are congruent. Discuss and show the proof. ∴ ∠2 ≅ ∠6 4. Subtraction Prop. 5. Def. of congruent angles. 1. Given:_____________ Prove: ___________ Figure:Teacher’s Note and Reminders Proof: Reason Statement Don’t Show the proof of the following. Forget! 2. Given: ∠1 ≅ ∠2 ∠3 ≅ ∠4 Prove: ∆COD ≅ ∆BOD 3. Given: OP ⊥ PE; EB ⊥ PE T is the midpoint of PE Prove: OP ≅ BE 365

WWhhaatt ttoo UUnnddeerrssttaanndd In this section, the discussion was about proofs. Go back to the previous section and compare your initial ideas with the iedeas This is a picture analysis activity, where students can form conditional statements discussed. How much of your initial ideas are found in the discussion? Which ideas are out of the picture. different and need revision? Teacher’s Note and Reminders Now that you know the important ideas about this topic, let’s go deeper by moving on to the next section. WWhhaatt ttoo UUnnddeerrssttaanndd What will you do before jumping into a conclusion? This is a picture analysis activity, where you can form conditional statements out of the picture. Proceed by forming another statement drawn from the original one. Activity 9 “CASE SOLVED” From the sets of geometric representations on real-life situations problems, write your reasons inductively or deductively on the \"Reason Out Activity Sheet\". Be ready to present your arguments to persuade your classmates. Your arguments will be rated in terms of coherence, mathematical thinking and conclusions made. Don’tForget! 366

In this section, the discussion was about proofs. Activity 10 SHALL WE MEET OR NOT?Go back to the previous section and compare your initial ideas with the The city’s newspaper will release its 2nd volume this year. It contains a new columndiscussion. How much of your initial ideas are found in the discussion? that will explain phenomena that deals with the concepts of mathematics. As a contributorWhich ideas are different and need revision? writer, you were tasked to write an article that will explain the “phenomenon of perspective” it shows that when two parallel lines are seen from a far, the lines intersect. Your article willNow that you know the important ideas about this topic, let’s go deeper by be evaluated by the head writer and editor-in-chief based on its coherence, mathematicalmoving on to the next section. thinking, and conclusions made. Teacher’s Note and Reminders In this section, the discussion was about proofs. What new realizations do you have about the topic? What new connections have you made for yourself? Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section. Don’t WWhhaatt ttooTTrraannssffeerrForget! In this section, you will be applying your understanding of inductive and deductive reasoning through the following activity that reflect meaningful and relevant situations. Activity 11 “IMBESTIGADOR” Math Magazine will release its November issue themed “MATH INVESTIGATES”. As one of the investigators you were tasked to make a mathematical investigation that will enlighten the readers by providing valid conclusions. The written output of your investigation will be presented to the Head writer, writers, and Editor-In-Chief and shall be evaluated based on to its coherence, clarity, judgment, and mathematical reasoning. In this section, your task was on math investigations. How did you find the performance task? How did the task help you see the application use of the topic? 367

Teacher’s Note and Reminders Don’t Forget! 368

TEACHING GUIDEModule 7: Triangle CongruenceA. Learning Outcomes Content Standard: The learner demonstrates understanding of the key concepts of triangle congruence. Performance Standard: The learner is able to communicate mathematical thinking with coherence and clarity, in formulating, investigating, analyzing and solving real life problems involving proving triangle congruence.UNPACKING THE STANDARDS FOR UNDERSTANDINGSUBJECT: LEARNING COMPETENCIESGrade 8 Mathematics 1. (K) Illustrates triangle congruenceQUARTER: 2. (K,S) States and illustrates the SAS, ASA, and SSS congruence postulatesThird Quarter 3. (S) Applies the postulates and theorems on triangle congruence to prove statementsSTRAND: on congruence including right trianglesGeometry 4. (S) Applies triangle congruence to geometric constructions of perpendicular andTOPIC: angle bisector.Triangle Congruence ESSENTIAL UNDERSTANDING: ESSENTIAL QUESTION:LESSONS:1. Definition of Congruent Triangles Students will understand that: How do you know that2. Congruence Postulates Triangle congruence can be proven deductively. the given triangles are3. Proving Congruence of Triangles congruent?4. Application of Triangle Congruence Real life problems requiring stability involve the use of triangle congruence. What ensures the stability of any structure? TRANSFER GOAL: Students will, on their own, solve real life problems pertaining to stability and structure, using triangle congruence. 369

B. Planning for AssessmentProduct/PerformanceThe following are products and performances that students are expected to accomplish with in this module. Assessment Map KNOWLEDGE UNDERSTANDING TRANSFER AND PROCESS/ (MEANING MAKING) TYPE Journal SKILLS Pre Test Rubric PRE-ASSESSMENT/ (ACQUISITION) Interpretation Journal Writing DIAGNOSTIC Pre – test Explanation Written Exercises Testing for congruence understandingFORMATIVE ASSESSMENT Interpretation Explanation Quiz Congruence Postulates Worksheet Interpretation SUMMATIVE Application ASSESSMENT ProvingSELF ASSESSMENT Post Test Interpretation Explanation Application Long Examination Interpretation Explanation Application 370

Assessment Matrix (Summative Test) Levels of What will I assess? How will I assess? How Will I Score?Assessment 1 point for every correctKnowledge The learner demonstrates understanding Paper and Pencil Test response 15% of key concepts of triangle congruence. Part I items 1, 2, 3, 12, 13Process/Skills State and illustrate the SAS, ASA, and 1 point for every correct 25% SSS congruence. response Prove triangle congruence Part II items 1 d-e, 3, 4, 1 point for every correct by SAS,ASA, and SSS. and 5 responseUnderstanding Apply triangle congruence to geometric Rubric for explanation 30% construction of perpendicular and angle Criteria: Clear bisector. Coherent Solve problems involving triangle Justified congruence. Rubric for drawing Criteria: Neat and Clear Accurate Justified Appropriate Relevant 371

The learner is able to formulate real-life Portfolio problems involving triangle congruence and solve these with utmost accuracy Cite two situations in real- using a variety of strategies. life where the concept is illustrated. Rubric on Problem Posing/ Formulation and Problem Formulate problems out Solving of these situations then Criteria: RelevantProduct solve them in as many Creative 30% ways as you can. Insightful Authentic GRASPS Assessment Clear Make a design or a sketch plan of a Rubric on Design/Sketch Plan suspension bridge. Apply Criteria: your understanding of the key concepts of triangle congruence.C. Planning for Teaching-LearningIntroduction: This module covers key concepts of triangle congruence. It is divided into four lessons, Lesson 1: Definition of congruent triangles Lesson 2: Congruence postulates Lesson 3: Proving congruence of triangles Lesson 4: Application of triangle congruence In all lessons, students are given the opportunity to use their prior knowledge and skills in learning triangle congruence.They are also given varied activities to process the knowledge and skills learned and reflect and further understand andtransfer their understanding of the different lessons. 372

As an introduction to the module, the teacher ask the following questions: o Have you ever wondered how bridges and buildings are designed? o What factors are being considered in making bridges and buildings? o How do problems on structure stability be solved? o When do you say triangles are congruent? o What ensures the stability of any structure? Entice the students to find out the answers to these questions and to determine the vast applications of triangle congruence in this module.Objectives: After the learners have gone through the lessons contained in this module, they are expected to: • Define and illustrate congruent triangles • State and illustrate the SAS, ASA and SSS Congruence Postulates; • Apply the postulates and theorems on triangle congruence to prove statements on congruence, including right triangles; and • Apply triangle congruence to geometric construction of perpendicular bisector and angle bisector.Learning Goals and Targets: In this lesson, students are expected to do the following: 1. Cite real-life situations where congruent triangles are illustrated and describe how these mathematics concepts are applied. 2. Define unfamiliar terms. 3. Explore the websites for better understanding of the lesson. 4. Write a journal on their experience in using the module. 5. Make a portfolio of their output. 6. Collaborate with their teacher and peers. 373

Pre-Assessment:1. In the figure ∆POG ≅ ∆SOR, what is the side corresponding to PO? a. OS PG b. RD O c. RS d. SO Answer: D RS2. Listed below are the six pairs of corresponding parts of congruent triangles. Name the congruent triangles. SA ≅ JO ∠D ≅ ∠Y AD ≅ OY ∠A ≅ ∠O SD ≅ JY ∠S ≅ ∠J a. ∆ASD ≅ ∆JOY b. ∆ADS ≅ ∆YJO c. ∆SAD ≅ ∆JOY d. ∆SAD ≅ ∆JYO Answer: A3. In ∆DOS, what side is included between ∠D and ∠O ? a. DO D b. DS SO c. SD d. SO Answer: A 374

4. Name the corresponding congruent parts as marked that will make each pair of triangles congruent by SAS. BR a. BY ≅ NR, ∠BOY ≅ ∠NOR, BO ≅ NO O b. BO ≅ NO, ∠BOY ≅ ∠NOR, RO ≅ YO c. YO ≅ OR, BO ≅ ON, ∠BOY ≅ ∠NOR d. ∠B ≅ ∠N,BO ≅ NO, OY ≅ OR Answer: B YN5. If corresponding congruent parts are marked, how can you prove ∆BEC ≅ ∆BAC? B a. ASA b. LL c. SAS d. SSS Answer: D C EA6. Identify the pairs of congruent right triangles and tell the congruence theorem used. M S a. ∆PMA ≅ ∆APS b. ∆MAP ≅ ∆SPA c. ∆MPA ≅ ∆SPA d. ∆AMP ≅ ∆PAS Answer: A AP 375

7. What property of congruence is illustrated in the statement? If AB ≅ DE, EF ≅ DE then AB ≅ EF. A. Symmetric C. Reflexive B. Transitive D. Multiplication Answer: B8. ∆GIV ≅ SAV deduce a statement about point V. G a. V is in the interior of ∆GIV. I V A b. V is in the exterior of ∆SAV. c. V is in the midpoint of GS. d. V is collinear with G and I. S Answer: C9. Is the statement “corresponding parts of congruent triangles are congruent” based on a. Definition c. Theorem b. Postulate d. Axiom Answer: B10. Use the marked triangles to write proper congruence statement. L OS C. LT ≅ MS OL ≅ ME A. LT ≅ MS OT ≅ SE LO ≅ ME ∆LOT ≅ ∆MSE OT ≅ MS ∆LOT ≅ ∆MES B. LT ≅ SM LO ≅ ME D. TL ≅ MS OT ≅ ES LO ≅ ME OT ≅ METE M ∆TOL ≅ ∆SME Answer: A 376

11. Hexagon CALDEZ has six congruent sides. C A CE, CD, CL are drawn on the hexagon L forming 4 triangles. Which triangles can you prove congruent? D a. ∆CEZ ≅ ∆CDE Z ∆CDE ≅ ∆CAL E b. ∆CEZ ≅ ∆CAL ∆CED ≅ ∆CLD c. ∆CED ≅ ∆CEZ ∆CLA ≅ ∆CLD d. ∆CZE ≅ ∆CED ∆DEC ≅ ∆LCD Answer: B12. ∆ABC ≅ ∆DEF, which segment is congruent to AB: a. BC b. AC c. DE d. EB Answer: C13. ∆SUM ≅ ∆PRO, which angle is congruent to ∠M? a. S b. P c. R d. O Answer: D 377

14. ∆TIN ≅ ∆CAN, then ∆NAC is congruent to ____. a. ∆ITN b. ∆NIT c. ∆TNI d. ∆INT Answer: B 15. Jancent knows that AB = XY and AC = XZ. What other information must he know to prove ∆ABC ≅ ∆XYZ by SAS postulate? a. ∠B ≅ ∠Y b. ∠C ≅ ∠Z c. ∠A ≅ ∠X Answer: C 16. Miguel knows that in ∆MIG and ∆JAN, MI = JA, IG = AN, and MG = JN. Which postulate or theorem can he use to prove the triangles congruent? a. ASA b. AAS c. ASA d. SSS Answer: D 17. In ∆ABC, AB = AC. If m∠B = 80, find the measure of ∠A. a. 20 b. 80 c. 100 d. 180 Answer: A 378

18. You are tasked to make a design of the flooring of a chapel using triangles. The available materials are square tiles. How are you going to make the design? a. Applying triangle congruence by ASA b. Applying triangle congruence by SAS. c. Applying triangle congruence by SSS d. Applying triangle congruence by AAS Answer: C For items 19 to 20Complete the proof. Choose the letter of the correct answer to fill the blank a. CO ≅ CO b. ASA c. SAS d. ∠BCO ≅ ∠ACO In ∆ABC, let O be a point in AB such that CO bisects ∠ACB, if AC ≅ BC.Prove that ∆ACO ≅ ∆BCO. Statements Reasons 1. AC ≅ BC 1. Given 2. CO bisects ∠ACB 2. Given Definition of angle bisector 3. ____(19)_____ 3. Reflexive Property of Congruence 4. CO ≅ CO ____(20)_______ 4. Answer: C 5. ∆ACD ≅ ∆BCO 5. Answer: D 379

WWhhaatt ttoo KKnnooww Lesson 1 Definition of Congruent Triangles Before doing Activity 1, Let the students answer the following questions on WWhhaatt ttoo KKnnooww their journal.Activating Prior Knowledge 1. What is the symbol for congruence? Let’s begin this lesson by finding out what congruent triangles are. As you go 2. If ∆ABC ≅ ∆XYZ, what are the six pairs of corresponding congruent parts? over the activities, keep on thinking “When are two triangles congruent?” 3. How do we measure an angle? 4. How can you draw an angle of specified measure? Activating Prior Knowledge 5. What is the sum of the measures of the angles of a triangle? 1. What is the symbol for congruence?For numbers six to ten define of Illustrate each of the following: 2. If ∆ABC ≅ ∆XYZ, what are the six pairs of corresponding congruent parts? 3. How do we measure an angle? 6. Midpoint 4. How can you draw an angle of specified measure? 7. Vertical angles 5. What is the sum of the measures of the angles of a triangle? 8. Right Triangle 9. Hypotenuse For numbers 6 to 10 define of Illustrate each of the following: 10. Isosceles Triangle 6. Midpoint 7. Vertical anglesMy idea of Congruent Triangles is ___________________________________ 8. Right Triangle______________________________________________________________ 9. Hypotenuse____________________________________________________ 10. Isosceles TriangleProvide the students opportunities to recall congruent figures. Define The wonders of Geometry are present everywhere, in nature and in structures.congruent triangles by doing activities 1 ad 2. Designs and patterns having the same size and same shape play important roles especially on the stability of buildings and bridges. What ensures the stability of any structures? Hook: In coming to school, have you met Polygon ? Name it and indicate where you met it. (Answers vary, Rectangles windows, 20 peso bill from my pocket triangles from bridges and buildings and houses etc.) 380

Provide the students opportunities to recall congruent figures. Definecongruent triangles by doing activities 1 and 2.Activity 1: Picture AnalysisShow the following pictures:From the picture shown, ask the students the given questions in Activity 1. Activity 1 PICTURE ANALYSISAnswers will be presented to the class and be discussed. From the picture shown, your group will answer the following questions: Teacher’s Note and Reminders 1. How will you relate the picture to your ambition? Don’t 2. If you were an architect or an engineer, what is your dream project? Forget! 3. What can you say about the long bridge in the picture? How about the tall building? (presence of congruent triangles, its stability, uses of bridges for economic progress, 4. Why are there triangles in the structures? Are the triangles congruent? When are two triangles congruent? 5. Why are bridges and buildings stable? Answers will be recorded. You gave your initial ideas on congruent triangles and the stability of bridges and buildings. Let’s now find out how others would answer the question and compare their ideas to our own, We will start by doing the next activity.! 381

WWhhaatt ttoo PPrroocceessss WWhhaatt ttoo PPrroocceessss In order to have an idea of congruent figures that will lead to the Let’s begin by finding out what congruent triangles are.definition of congruent triangles; let the student perform activity 2. See toit that everybody participates. The teacher is allowed to play music as they Activity 2 FIND YOUR PARTNERperform the activity. Let us see your knowledge about congruent fi. Teacher’s Note and Reminders Instruction Don’t Your group (with 10 members) will be given five pairs of congruent figures, each Forget! shape for each member. At the count of three, find your partner by matching the shape that you have with another’s shape. QU ?E S T I ONS 1. Why/How did you choose your partner? 2. Describe the two figures you have. 3. What can you say about the size and shape of the two figures? 4. We say that congruent figures have the same size and the same shape. Verify that you have congruent figures. For each group you pick up a pair of congruent triangles BE A CF D Name your triangles as ∆ABC and ∆DEF as shown in the figure.After the students have chosen and named the two triangles, explain to them Investigate: Matching vertices of the two trianglesthe investigation they will make that will lead them to the formal definition of First Match: ABC ↔ EDF (A corresponds to E, B corresponds to D,congruent triangles. Students will fill up the activity sheet. C corresponds to F) Second Match: ABC ↔ EFD Third Match: ABC ↔ DEF In which of the above pairings are the two triangles congruent? Fill up the activity sheet on the next page. 382

Let the students state the definition of congruent triangles based on the Group No.__________investigation made. Allow them to answer the following questions on theirjournal: Match Corresponding Congruent Corresponding Congruent• In what pairing or match do the two triangles coincide? sides or not Angles Or not• What are congruent triangles? First• How many pairs of corresponding parts are congruent? Second congruent? congruent?• Illustrate ∆TNX ≅ ∆HOP. Put identical markings on congruent Third corresponding parts. Two triangles are congruent if their vertices can be paired so that corresponding• Where do you see congruent triangles? sides are congruent and corresponding angles are congruent.Ask the students to answer Exercise 1. ∆ABC ≅ ∆DEF Read as \"triangle ABC is congruent to triangle DEF.\" Teacher’s Note and Reminders ≅ symbol for congruency ∆ symbol for triangle. The congruent corresponding parts are marked identically. Can you name the corresponding congruent sides? Corresponding congruent angles? Answer the questions below in your journal:  What are congruent triangles?  How many pairs of corresponding parts are congruent if two triangles are congruent?  Illustrate ∆TNX ≅ ∆HOP Put identical markings on congruent corresponding parts.  Where do you see congruent triangles? Don’t Exercise 1 A CForget! B D 1. ∆ABD ≅ ∆CBD, Write down the six pairs of congruent corresponding parts 2. Which triangles are congruent if MA ≅ KF, AX ≅ FC, MX ≅ KC; ∠M ≅ ∠K, ∠A ≅ ∠F, ∠X ≅ ∠C. Draw the triangles. 383

Challenge the students to prove triangle congruence by showing lesser 3. Which of the following shows the correct congruence statement for the figurenumber of corresponding congruent parts. below?Discuss first with the students the parts of a triangle in terms of relativeposition then let them answer Exercise 2 and do Activities 3-5. a. ∆PQR ≅ ∆KJL b. ∆PQR ≅ ∆LJK Teacher’s Note and Reminders c. ∆PQR ≅ ∆LKJ d. ∆PQR ≅ ∆JLK You can now define what congruent triangles are .In order to say that the two triangles are congruent, we must show that all six pairs of corresponding parts of the two triangles are congruent. Let us see how can we verify two triangles congruent using fewer pairs of congruent corresponding parts. Topic 2: Triangle Congruence Postulates Before we study the postulates that give some ways to show that the two triangles are congruent given less number of corresponding congruent parts, let us first identify the parts of a triangle in terms of their relative positions. Included angle is the angle between two sides of a triangle. Included side is the side common to two angles of a triangle. Don’t In ∆SON S ∠S is included between SN and SO.Forget! ∠O is included between OS and ON. ∠N is included between NS and NO. SO is included between ∠S and ∠O. ON is included between ∠O and ∠N. O SN is included between ∠S and ∠N. N 384

Always have a drill and review identifying included side, included angle Exercise 2before you ask the students to do the activities. Given ∆FOR, can you answer the following questions even without the figure?From Activity 3, let the students make their own generalization on SAS 1. What is the included angle between FO and OR?congruence postulate. Ask them to answer Exercise 3 2. What is the Included angle between FR and FO? 3. What is the included angle between FR and RO? Teacher’s Note and Reminders 4. What is the included side between ∠F and ∠R? 5. What is the included side between ∠O and ∠R? 6. What is the included side between ∠F and ∠O? Activity 3 LESS IS MORE Don’t SAS (Side-Angle-Side) Congruence PostulateForget! 1. Prepare a ruler, a protractor ,a pencil and a bond paper. 2. Work in group of four. 3. Follow the demonstration by the teacher. a. Draw a 7- inch segment. b. Name it BE. c. Using your protractor make angle B equal to 70o degrees. d. From the vertex draw BL measuring 8 inches long. e. How many triangles can be formed? f. Draw ∆BEL g. Compare your triangle with the triangles of the other members of the group. Do you have congruent triangles? h. Lay one triangle on top of the others. Are all the corresponding sides congruent? How about the corresponding angles? i. What can you say about any pair of congruent triangles ? SAS (Side-Angle-Side) Congruence Postulate If the two sides and an included angle of one triangle are congruent to the corresponding two sides and the included angle of another triangle, then the triangles are congruent. If MA ≅ TI, ∠M ≅ ∠T, MR ≅ TN Then ∆MAR ≅ ∆TIN by SAS Congruence Postulate Mark the congruent parts. AI M RT N 385

Let the student state the generalization on ASA congruence, deduced from Exercise 3the activity they have just performed. Complete the congruence statement using the SAS congruence postulate. Teacher’s Note and Reminders 1. ∆BIG ≅ ∆_____ 3. ∆ABO ≅ ∆_____ I A B AO D B GF T C 2. ∆PON ≅ ∆____ 4. ∆PAT ≅ ∆_____ N A D O E P P TS S After showing that the two triangles are congruent showing only two sides and the included angle of one triangle to two sides and included angle of another triangle, you try another way by doing activity 4 Activity 4 TRY MORE Don’t ASA (Angle-Side Angle) CongruenceForget! Prepare the following materials; pencil, ruler, protractor, a pair of scissors Working independently, use a ruler and a protractor to draw ∆BOY with two angles and the included side having the following measures: m∠B = 50, m∠O = 70 and BO =18 cm 1. Draw BO measuring 18 cm 2. With B as vertex, draw angle B measuring 50. 3. With O as vertex, draw angle O measuring 70. 4. Name the intersection as Y. 5. Cut out the triangle and compare it with four of your classmates. 6. Describe the triangles. 7. Put identical marks on the congruent corresponding sides and angles. 8. Identify the parts of the triangles which are given congruent. 386

From the Activity 5, ask the students to state their own generalization on ASA (Angle-Side-Angle) Congruence PostulateSSS Congruence postulate. If the two angles and the included side of one triangle are congruent to the Let them answer Exercise 4. corresponding two angles and an included side of another triangle, then the triangles are congruent. Teacher’s Note and Reminders If ∠A ≅ ∠E, JA ≅ ME, ∠J ≅ ∠M, then ∆JAY ≅ ∆MEL Draw the triangles and mark the congruent parts. Activity 5 SIDE UP SSS (Side-Side-Side) Congruence Postulate You need patty papers, pencil , a pair of scissors 1. Draw a large scalene triangle on your patty paper. 2. Copy the three sides separately onto another patty paper and mark a dot at each endpoint. Cut the patty paper into three strips with one side on each strip. 3. Arrange the three segments into a triangle by placing one endpoint on top of the another. Don’t 4. With a third patty paper, trace the triangleForget! formed. Compare the new triangle with the original triangle. Are they congruent? 5. Try rearranging the three segments into another triangle. Can you make a triangle not congruent to the original triangle? Compare your results with the results of others near you. 387

Now that the students can show triangles congruent with SSS (Side-Side-Side) Congruence Postulate• two corresponding sides and an included angle If the three sides of one triangle are congruent to the three sides of another• two angles and an included side triangle, then the triangles are congruent.• three pairs of corresponding sides congruent, they are now ready to If EC ≅ BP, ES ≅ BJ, CS ≅ PJ, then ∆ESC ≅ ∆BJP, draw the triangles and mark the prove two triangles congruent deductively. congruent parts., then answer exercise 4. Teacher’s Note and Reminders Exercise 4 Corresponding congruent parts are marked. Indicate the additional corresponding parts needed to make the triangles congruent by using the specified congruence postulates. A D a. ASA _______ b. SAS _______ C BF E P . L O a. SAS ______ M b. SSS ______ T Don’t a. SAS ______Forget! b. ASA ______ Now that you can show triangles congruent with • two corresponding sides and an included angle • two angles and an included side • three pairs of corresponding sides congruent, you are now ready to prove two triangles congruent deductively. 388