Mathematics Grade 8 Part 2

Allow students to study and discuss with their group the proof in No.1 then Topic 3: Proving Triangle Congruencelater let them complete the proof in no. 2. Do Activity 6 Let’s Do IT. Activity 6 LET’S DO ITTell the students that proving triangle congruence will lead them to provemore theorems that will be helpful for them in the next lesson. Let’s find out how we can apply the Congruence Postulates to prove two triangles Teacher’s Note and Reminders congruent. Study the following example and answer exercise 5. Given: AB ≅ DE B CF E ∠B ≅ ∠E A D BC ≅ EF Prove: ∆ABC ≅ ∆DEF 1. AB ≅ DE Reasons 2. ∠B ≅ ∠E 1. Given 3. BC ≅ EF 2. Given 4. ∆ABC ≅ ∆DEF 3. Given 4. SAS Postulate Exercise 5 E Try this B Given: BE ≅ LO, BO ≅ LE Prove: ∆BEL ≅ ∆LOB L O Complete the Proof: Don’t Statements ReasonsForget! 1. 1. Given 2. BO ≅ LE 2. 3. 3 4. 4. ∆BEL ≅ ∆LOB 389

Discuss this with the students by showing more examples. Let’s try to prove a theorem on congruence,Let’s try to prove a theorem on congruence, Given the triangles below, a pair of corresponding sides are congruent, and two pairs Given the triangles below, a pair of corresponding sides are congruent, and two of corresponding angles have the same measure.pairs of corresponding angles have the same measure. D DC C 47 47 47 47 O 48 AO 48 A G T 48 G 48 T Work in Pairs to discuss the proof of the theorem by completing the flow chart Work in Pairs to discuss the proof of the theorem by completing the flow chart ∠D ≅ ∠C ∠G ≈ _____∠D ≅ ∠C ∠G ≈ _____ ∠O ≅ _____ ∠O ≅ _____ ∆DOG ≅ _____ OG ≅ AT ∆DOG ≅ _____ OG ≅ AT Supply the reason for each Supply the reason for eachLet the students work in pairs to discuss the proof of the theorem by When you completed the proof, review the parts of the two triangles which arecompleting the flow chart. given congruent. Have you realized that you have just proved the AAS congruence Theorem? 390

Tell the students that when they completed the proof review what parts of AAS (Angle-Angle-Side) Congruence Theoremthe triangles are shown congruent. Let them realized that they have just If two angles and a non-included side of one triangle are congruent to theshown the proof of a theorem on congruence, the AAS congruence theorem. corresponding two angles and a non-included side of another triangle, then the triangles are congruent.Ask the students to study the example. Tell them that with the use of SAS,ASA, Example: SSS postulates and AAS congruence theorem, they can now prove two Given: ∠NER ≅ ∠NVRtriangles congruent, segments and angles congruent that can lead to prove RN bisects ∠ERVmore theorems especially on right triangles Let them do activity 7. Prove: ∆ENR ≅ ∆VNR Teacher’s Note and Reminders Statements Reasons 1. ∠NER ≅ ∠NVR 2. RN bisects ∠ERV 1. Given 3. ∠NER ≅ ∠NVR 2. Given 4. RN ≅ RN 3. Definition of angle bisector 5. ∆ENR ≅ ∆VNR 4. Reflexive Property 5. AAS Postulate Exercise 6 Complete the congruence statement by AAS congruence. Figure Congruence Statement ∆BOX ≅ ________ Don’tForget! ∆GAS ≅ _________ 391

Before you ask the students to do Activity 7, tell them that within their group, ∆FED ≅ ________they will recall the parts of a right triangle by performing this activity. Teacher’s Note and Reminders Don’t ∆BAM ≅ ________Forget! CM bisects BL at A ∠L ≅ ∠B How are we going to apply the congruence postulates and theorem In right triangles? Let us now consider the test for proving two right triangles congruent. Activity 7 KEEP RIGHT Recall the parts of a right triangle with your groupmates . 1. Get a rectangular sheet of paper. 2. Divide the rectangle along a diagonal. 3. Discuss with your group and illustrate the the sides and angles of a right triangle using your cut outs • What do you call the side opposite the right angle? • What do you call the perpendicular sides? • How many acute angles are there in a right triangle? 4. Name your triangles as shown below BS I GM L 392

Ask each group to present the proof to the class using a two column form, a 5 . If ∆BIG and ∆SML are right triangles, ∠I and ∠M are right, BI ≅ SM, IG ≅ MLflow chart, or in paragraph form to deduce the theorem. prove ∆BIG ≅ ∆SML.Now give the students a chance to prove on their own the next theorems on 6. Discuss the proof with your group.right triangles congruence.Ask each group to make a power-point presentation using a flowchart to 7. Answer the following questions:prove the following theorems. • What kind of triangles did you prove congruent? • What parts of the right triangles are given congruent? Teacher’s Note and Reminders • Complete the statement: If the ______ of one right triangle are congruent to the corresponding ___ of another right triangle, then the triangles are _____. Since all right angles are congruent you can now use only two pairs of corresponding parts congruent in order to prove two triangles congruent, The proof you have shown is the proof of the LL Congruence Theorem . LL Congruence Theorem If the legs of one right triangle are congruent to the legs of another right triangle, then the triangles are congruent. The LL Congruence Theorem was deduced from SAS Congruence Postulate. Consider the right triangles HOT and DAY with right angles at O and A, respectively, such that HO ≅ DA, and ∠H ≅ ∠D. HD Prove: ∆HOT ≅ ∆DAY. Don’t T OA YForget! Each group will present the proof to the class either by two column form or using flow chart or paragraph form to deduce the theorem: LA (leg-acute angle) Congruence Theorem If a leg and an acute angle of one right triangle are congruent to a leg and an acute angle of another right triangle, then the triangles are congruent. Now it’s your turn to prove the other two theorems on the congruence of right triangles. 393

Allow students to present their work the next day. After the discussion, let Activity 8 IT’S MY TURNthem answer Exercise 7.Teacher’s Note and Reminders Each group will make a power point presentation using flowchart to prove the following theorems. HyL (Hypotenuse-leg) Congruence Theorem If the hypotenuse and a leg of one right triangle are congruent to the corresponding hypotenuse and a leg of another triangle, then the triangles are congruent. HyA (Hypotenuse-Acute angle) Congruence Theorem If the hypotenuse and an acute angle of one right triangle are congruent to the corresponding hypotenuse and an acute angle of another right triangle, then the triangles are congruent. Guide: 1. Draw the figure 2. What is given and what is to be proved? 3. Write the proof in two-column form. Exercise 7 In each figure, congruent parts are marked. Give additional congruent parts to prove that the right triangles are congruent and state the congruence theorem that .justifies your answer. 1. A 2. B 3A. B F 2 11 A 2C C DE Don’t B_________D_________C_ _________D__________ ___________________Forget! ___________________ ___________________ ___________________ 394

After studying the congruence postulates and theorems the students are State a congruence theorem. on right triangles.now ready to apply them. How can they prove that the two angles or two 4. __________ 5. __________segments are congruent?Ask them to perform Activity 9. Teacher’s Note and Reminders 6. 7. 8. Topic 4: Application of Triangle Congruence After studying the congruence postulates and theorems you are now ready to apply them. How can you prove that two angles or two segments are congruent? If they are parts of congruent triangles we can conclude that they are congruent. Let us see how. Don’t Activity 9 WHAT ELSE? VertexForget! Angle Do you still remember what an isosceles triangle is? leg leg A triangle is isosceles if two of its sides are Base Angle congruent. The congruent sides are its legs. the third side is the base, the angles opposite the congruent sides are the base angles and the angle included by Base the legs is the vertex angle. Angle 395

After the discussion on parts of isosceles triangle, Ask the students to Consider ∆TMY with TM ≅ TYcomplete the proof of the exercise below. Is ∠M ≅ ∠Y? You find out by completing the proof. Consider ∆TMY with TM ≅ TY Is ∠M ≅ ∠Y? Remember that if they are corresponding parts of congruent triangles then they are You find out by completing the proof. congruent. T Remember that if they are corresponding parts of congruent triangles then 1. Draw the bisector TO of ∠T which intersects MY at O.they are congruent. 2. _______ ≅ ________ by definition of a bisector 3. _______ ≅ ________ given T 4. _______ ≅ ________ (Why) _____________ 1. Draw the bisector TO of ∠T which intersects MY at O. O 5. _______ ≅ ________ SAS 2. _______ ≅ ________ by definition of a bisector 6. ∠M ≅ ∠Y __________________ 3. _______ ≅ ________ given MOY 4. _______ ≅ ________ (Why) _____________ Isosceles Triangle Theorem: 5. _______ ≅ ________ SAS If two sides of a triangle are congruent then the angles opposite these sides are 6. ∠M ≅ ∠Y __________________ M Y congruent.As an exercise, ask the student to prove the converse of isosceles triangle How about the converse of isosceles triangle theorem: If two angles of a triangletheorem: are congruent then the sides opposite these angles are congruent.Challenge the students to prove that: An equilateral triangle is equiangular. Exercise 8They will be guided by the figure and questions below. Let them answer Prove the converse of the isosceles triangle theorem with your group.Exercise . Discuss with your group the proof of the statement: An equilateral triangle isGuide the students to prove: The bisector of the vertex angle of an Misosceles triangle is perpendicular to the base at its midpoint. Follow equiangular.the suggested procedure. The students can do this in group and output willbe presented to the class. Use the figure and be guided by the questions below. Given: ∆MIS is equilateral Prove: ∆MIS is equiangularTeacher’s Note and Reminders In order to prove that ∆MIS is equiangular you must prove first that ∠M ≅ ∠I ≅ ∠S Don’t Forget! 1. MI ≅ MS Why? 2. What kind of triangle is ∆MIS?. 3. What angles are congruent? Why? 4. MI ≅ MS Why 5. What angles are congruent? Why? 6 ∠M ≅ ∠I ≅ ∠S Why? How will you show that each angle of an equilateral triangle measures 60°? Guide Questions: a. What is the sum of the measures of the angles of a triangle? b. What is true about equilateral triangle? 396

Guide the students to prove the following: The bisector of the vertex angle Exercise 9of an isosceles triangle is perpendicular to the base at its midpoint.Follow the suggested procedure. The students can do this in group and 1. What is the difference between an equilateral triangle and isosceles triangle?output will be presented to the class. 2. One angle of an isosceles triangle measures 60°. What are the measures ofProcedure: the other two angles? a. Draw an Isosceles ∆ABC. 3. An angle of an isosceles triangle is 50°. What are the measures of the other b. Draw the bisector BE of the vertex ∠B which intersects AC at E. c. Prove that the two triangles BEA and BEC are congruent. two angles? Is there another possible triangle? d. Show that E is the midpoint AC. Discuss the proof of: The bisector of the vertex angle of an isosceles triangle is e. Show BE is perpendicular to AC at E. (Remember that segments perpendicular to the base at its midpoint. Do this with your group. are perpendicular if they form right angles.) Procedure:In this section, discussion was on Congruent Triangles. Go back the previoussection and compare your initial ideas with the discussion. How much of your a. Draw an Isosceles ∆ABC.initial ideas are found in the discussion? Which ideas are different and need b. Draw the bisector BE of the vertex ∠B which intersects AC at E.revision? c. Prove that the two triangles BEA and BEC are congruent.Now that you know the important ideas about this topic, let's go deeper bymoving on the next section. d. Show that E is the midpoint AC.Your goal in this section is to take a closer look at some aspects of the topic. e. Show BE is perpendicular to AC at E. (Remember that segments areAnd keep in mind the question: How does knowledge in trianglecongruence will help you to solve real-life problems? perpendicular if they form right angles.)WWhhaatt ttoo UUnnddeerrssttaanndd Your work will be presented in class. Theorem: The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint. In this section, the discussion was on Congruent Triangles Go back to the previous section and compare your initial ideas with the discussion. How much of your initial ideas are found in the discussion? Which ideas are different and need revision? Now that you know the important ideas about this topic, let’s go deeper by moving on to the next section. Ask the student to reflect on the different lessons tackled in the module by WWhhaatt ttoo UUnnddeerrssttaannddanswering the questions. Your goal in this section is to take a closer look at some aspects of the topic. And keep in your mind the question: “How does knowledge in triangle congruence will help you to solve real life problems?” 397

Students can write their answers in their journal, then do Activity 10 Questions: • When are two triangles congruent? Teacher’s Note and Reminders • What are the conditions for triangle congruence.? • How can we show congruent triangles through paper folding? Don’t • Say something about Isosceles triangle. Forget! • Is equilateral triangle isosceles? • Is equilateral triangle equiangular? • What can you say about the bisector of the vertex angle of an isosceles triangle? Activity 10 FLY FLY FLY During the Math Fair, one of the activities is a symposium in which the delegates will report on an inquiry about an important concept in Math. You will report on how congruent triangles are applied in real-life. Your query revolves around this situation; 1. Design at most 5 different paper planes using congruent triangles. 2. Let it fly and record the flying time and compare which one is the most stable. 3. Point out the factors that affect the stability of the plane. 4. Explain why such principle works. Procedure: 5. Draw out conclusion and make recommendations. 1. Each group will prepare 5 paper planes 2. Apply your knowledge on triangle congruence. 3. Follow steps 2 to 5. 4. What is the importance of congruent triangles in making paper planes? Activity 11 SARANGOLA NI PEPE Another application of congruent triangles is on stability of your kites. Show us how triangle congruence works. In the upcoming City Festival, there will be a kite flying. You are to submit a certain design of kite and an instruction guide of how it operates. The designer who can come up with a kite which can fly the longest wins a prize. Present the mechanics on how you come up with such a design. 398

Another challenge to your students is this task which they can do at home. Activity 12 3 – 2 – 1 CARDSubmit a journal on how you proved two triangles are congruent. Since you are done with the concepts and activities about triangle congruence, nowDid you enjoy these lesson on triangle congruence? let us summarize it by completing the table below:Take a picture of triangles in your house. Identify how each of these 3 things you have learnedcongruences could help a builder construct furnitures. Make a portfolio of these 2 things which are interestingpictures and discussion. 1 question you still have in mind Teacher’s Note and Reminders WWhhaatt ttooTTrraannssffeerr Don’t PerformanceForget! GRASPS TASK (S) One of the moves of the City Council for economic development is to connect a nearby island to the mainland with a suspension bridge for easy accessibility of the people. Those from the island can deliver their produce and those from the mainland can enjoy the beautiful scenery and beaches of the island. (R) As one of the engineers of the DPWH who is commissioned by the Special Project Committee, (G) you are tasked to present (P) a design/blueprint (P) of a suspension bridge to the (A) City Council together with the City Engineers. (S) Your presentation will be evaluated according to its accuracy, practicality, stability and mathematical reasoning. 399

Teacher’s Note and Reminders Now that you are done check your work with the rubric below. Don’t CRITERIA Outstanding Satisfactory Developing Beginning RATING Forget! 4 3 2 1 Accuracy The The Some The computations computations computations computations are accurate are accurate are erroneous are erroneous and show a and show and show the and do not wise use of the use of use of some show the use the concepts the concepts concepts of the concepts of triangle of triangle of triangle of triangle congruence. congruence. congruence. congruence. The design is The design is The design The design comprehensive and displays presentable and makes use of doesn’t use the aesthetic aspects of the makes use of the geometric geometric mathematical Creativity concepts the concepts representations representations learned. of geometric but not and not representations. presentable. presentable. The design The design The design The design is stable, is stable, makes use of does not use comprehensive presentable triangles but triangles and is and displays and makes use not stable. not stable. the aesthetic of congruent Stability aspects of triangles. the principles of triangle congruence. Mathematical The The explanation The The reasoning explanation is clear and explanation is explanation is is clear, coherent. It understandable incomplete and exhaustive covers the but not logical. inconsistent. or thorough important and coherent. concepts. It includes interesting facts and principles. OVERALL RATING Another challenge to you is this task for you to accomplish at home Submit a journal on how you proved two triangles congruent. Did you enjoy the lesson on triangle congruence? Take a picture of triangles in the house. Identify how each of these congruences could help a builder to construct a furniture. Make a portfolio of these pictures and discussion. 400

Teacher’s Note and Reminders SUMMARY Don’t Designs and patterns having the same size and the same shape are seen in almost Forget! all places. You can see them in bridges, buildings, towers, in furniture even in handicrafts and fabrics Congruence of triangles has many applications in real world. Architects and engineers use triangles when they build structures because they are considered to be the most stable of all geometric figures. Triangles are oftentimes used as frameworks, supports for many construction works. They need to be congruent. In this module you have learned that: • Two triangles are congruent if their vertices can be paired such that corresponding sides are congruent and corresponding angles are congruent. • The three postulates for triangle congruence are: a. SAS Congruence – if two sides and the included angle of one triangle are congruent respectively two sides and the included angle of another triangle then the triangles are congruent. b. ASA Congruence – if two angles and the included side of one triangle are congruent respectively two angles and the included side of another triangle then the triangles are congruent. c. SSS Congruence – if the three sides of one triangle are congruent respectively three sides of another triangles then the triangles are congruent. • AAS Congruence Theorem – if the two angles and the non-included side of one triangle are congruent to the two angles and the non-included side of another triangle than the triangles are congruent. • The congruence theorems for right triangles are: a. LL Congruence – if the legs of one right triangle are congruent respectively to the legs of another right triangle, then the triangles are congruent. b. LA Congruence – if a leg and an acute angle of one triangle are congruent respectively to a leg and an acute angle of another right triangle, then the triangles are congruent. c. HyL Congruence – if the hypotenuse and a leg of one right triangle are congruent respectively to the hypotenuse and a leg of another right triangle, the triangles are congruent. d. HyA Congruence – if the hypotenuse and an acute angle of one right triangle are congruent respectively to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent. • Isosceles Triangle Theorem – If two sides of a triangle are congruent then the angles opposite these sides are congruent. • Converse of Isosceles Triangle Theorem – if two angles of a triangle are congruent then the sides opposite these angles are congruent. • An equilateral triangle is equiangular. • The measure of each angle of an equilateral triangle is 60°. 401

Teacher’s Note and Reminders Don’t Forget! 402

TEACHING GUIDEModule 8: Inequalities in TrianglesA. Learning Outcomes All activities and inputs in this module that you have to facilitate are aligned with the content and performance standards of the K to 12 Mathematics Curriculum for Grade 8. Ensuring that students undertake all the activities at the specified time with your maximum technical assistance lies under your care. The table below shows how the standards are unpacked. UNPACKING THE STANDARDS FOR UNDERSTANDINGSUBJECT: LEARNING COMPETENCIESGrade 8 • (KNOWLEDGE) State and illustrate the theorems on triangle inequalities such as exterior angle inequalityMathematics theorem, triangle inequality theorems, hinge theorem and its converse.QUARTER: • (SKILL) Apply theorems on triangle inequalities to:Third Quarter a. determine possible measures for the angles and sides of triangles. b. justify claims about the unequal relationships between side and angle measures.STRAND:Geometry • (SKILL) Use the theorems on inequalities in triangles to prove statements involving triangle inequalities.TOPIC: ESSENTIAL UNDERSTANDING: ESSENTIAL QUESTION:Inequalities in Students will understand that inequalities in triangles can be How can you justify inequalities in trian-Triangles justified deductively. gles?LESSON:Inequalities in TRANSFER GOAL: Students will on their own justify inequalities in triangles that are evident in the things around us such as inTriangles artworks and designs. 403

B. Planning for Assessment To assess learning, students should perform a task to demonstrate their understanding of Inequalities in Triangles. It is expected that students, having been equipped with knowledge and skills on inequalities in triangles, would come up with a product—a design and a miniature model of a folding ladder that can reach as high as 10 feet. This task is found in Activity No. 23 of the module Assessment Map To ensure understanding and learning, students should be engaged in different learning experiences with corresponding assessment. The table below shows the assessment at different stages of the learning process. Details of this assessment map will guide you which items in each stage of assessment are under specific domains—Knowledge, Process/Skills, Understanding or Performance. . Be sure to expose students to varied assessment in this module in order to develop their critical thinking and problem solving skills.TYPE KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PERFORMANCEPre – assessment/ • Pre-Test Items • Pretest Items • Pretest Items • Pretest Items Diagnostic No. 1,2 and 10 No. 3, 4, 7, 12, and 13 No. 5, 6, 8, 9, 11, No. 14-20 and 14 404

Revisiting and Modifying Revisiting and Revisiting and • Answering Questions of Answers in Activity No. Modifying Answers in Modifying Answers the following activities: 1 Activity No. 3 in Activity No. 2 Act. Items 8 1-2 • Quiz • Completing the tables • Answering 9 8-9 Quiz 10 5 1 Items of the following Questions of 21 1-11 2 A 1-3 3 A 1-3 activities: 4, 5, 6, 7, the following • Answering Quiz Items 4 A 1, 5 Quiz Items 9, 10 activities: 1 A 1-3 1 2C • Answering Questions Act. Items 3 9-12 of the following 5 E1 activities: 4 1-5 • Answering Questions: Act. Items 5 1-5, 8  Mathematics in the 46 74 Kitchen  Mathematics in Art: 5 6-7 9 1-8 Geometric Shapes forFormative 6 1-2 10 1-4 Foundation Piecing  Mathematics for Eco- 7 1-3 22 1-3 Architecture • Completing the • Answering Quiz  Mathematics in the Garden proofs of the following items activities: No. 11, 12, Quiz Items 13, 14, 15, 16 1C • Answering Quiz items 3 A 6-11 Quiz Items 4 2-3 1B 5 A 1-4, 2B B 1-5, 3 A 2-4 D 5 C 1-4 405

• Answering  Mathematics in the Questions Geography about Watch- this problems  Mathematics in in the following Architecture: The World’s activities: 17, 18, Thinnest House 19, 20  Mathematics in Art: Color • Solving It’s-Your- Triangle Turn problems of the following  Mathematics in activities: 17, 18, Psychology 19, 20  Mathematics in Fashion  Career in Mathematics: Air Traffic Controller TYPE KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PERFORMANCESummative • Post-Test Items • Pretest Items • Pretest Items • Pretest Items No. 1,2 & 10 No. 3, 4, 7, 12, & 13 No. 5, 6, 8, 9, 11, No. 14-20 Finalizing Answers in Finalizing Answers in Activity No. 1 Activity No. 3 & 14 Finalizing Answers Act 23: Creation of a design and in Activity No. 2 a miniature model of a folding ladder that can reach as high as 10 feet—allowing its user to gain access to their ceilings/ roofs during floods caused by typhoons or monsoon rains. All items in Activity The standards are as follows: No. 24  Product must be efficient,  The design must be creative,  Measurements are accurate  Mathematical Justification of the design is logically clear, convincing, and professionally delivered 406

TYPE KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PERFORMANCESelf - assessment Answering Activity No. 3 Answering Activity Answering questions in More Answering Activity No. 1 No. 2 Triangular Designs and Artworks Assessment Matrix (Summative Test) Post-Test Items by Levels of Assessment Knowledge Process/ Skills Understanding ProductWhat will I assess? 3 items 5 items 6 items 6 items 15% 25% 30% 30% Scoring: One point EachCompetency No. 1: State and illustrate the theorems 1, 2on triangle inequalities such as exterior angle inequality 20theoremCompetency No. 2: Apply theorems on triangle inequalities 4, 11, 13to determine possible measures for the angles and sides of 5, 10triangles. 15, 16Competency No. 3: Apply theorems on triangle inequalities 3to justify claims about the unequal relationships between 12, 14side and angle measures. 6, 8 17Competency No. 4: Use the theorems on inequalities in 7, 9triangles to prove statements involving triangle inequalities 18, 19Competency Nos. 1, 2, 3, and 4 Activity: Creation of a Design or Product Scoring: By Rubrics 407

C. Planning for Teaching-Learning Introduction: The unit lesson on Geometry for Grade 8 is to be delivered in the Third Quarter of the school year. Triangle Inequalities is the third chapter of Geometry for Grade 8. Since there are four chapters in this unit, you are expected to facilitate this lesson within 15 days, non-inclusive of extra time student spend for tasks that you may most likely assign for students to do in their independent/cooperative learning time, free time, or after school. INTRODUCTION AND FOCUS QUESTIONS: Aside from arresting the attention and interest of the students, the introduction stresses the purpose of studying inequalities in triangles. The introduction, through the essential question, serves as a steering mechanism of the lesson. All sections and activities in the lesson are geared towards the goal of answering it. As the learning facilitator, your role is to emphasize the Essential Question in the introduction and to remind the students about it in every section of the module. Your key role is to underscore that the process of answering the essential question on how inequalities in triangles can be justified will: • improve their attention to details; • shape their deductive thinking; • hone their reasoning skills; and • polish their mathematical communication. . 408

LESSONS AND COVERAGE: This section of the learning module cites the subtopics of Inequalities in Triangles and the competencies that will becovered in the module. Your task is to know these competencies so you can ensure that students shall have learned them at theend of the lesson.MODULE MAP: Through the Module Map, you will be able to show to the students that • inequalities exist in one triangle and in two triangles • four theorems can be developed, verified, and proved regarding inequalities in one triangle • two theorems can be developed, verified, and proved regarding inequalities in two trianglesPRE-ASSESSMENT: This section features the test that diagnoses what students already know about the topic before the actual teaching of thelesson. This feedback information is valuable to you because it directs you on how to proceed as a facilitator of learning. As aresult, you are able to provide the appropriate technical assistance students need as the lesson unfolds. 409

Answer Key to Pre-Test III. PRE - ASSESSMENT1. C The measure of an exterior angle of a triangle is always Find out how much you already know about this topic. On a separate sheet, greater than either remote interior angle. Basis: Exterior write only the letter of the choice that you think best answers the question. Please Angle Inequality Theorem. answer all items. During the checking, take note of the items that you were not able to answer correctly and find out the right answers as you go through this module.2. B Angle 5 is an exterior angle of triangle TYP because segment PR is an extension of side TP. Basis: Definition of 1. The measure of an exterior angle of a triangle is always ____________. Exterior Angle. a. greater than its adjacent interior angle.3. B Marie was not able to form a triangle because the sum b. less than its adjacent interior angle. of the two shorter lengths 4 and 5 is not greater than the c. greater than either remote interior angle. third side of 9 inches. Basis: Triangle Inequality Theorem d. less than either remote interior angle. 3 (S1 + S2 > S3). 2. Which of the following angles is an exterior angle of ∆TYP?4. A Working Inequality 4x − 3 < 42o Basis: Triangle Inequality Theorem 3 (S1 + S2 > S3) UT5. B Basis: Converse of Hinge Theorem6. B The included angle between two distances 3 km and 4 km 71 P 2 covered by Oliver is 150o. This is larger than that of Ruel's--140o. 5 Therefore, Oliver is father because his distance is opposite a larger angle. Basis: Hinge Theorem. 4 37. B 68. C Basis: Converse of Hinge Theorem.9. B Conclusions must be based on complete facts. Y R10. A m∠D = 180 − (86 + 49) = 45. The shortest side is ∠D. Therefore the shortest side is opposite it. Basis: Triangle A. ∠4 B. ∠5 C. ∠6 D. ∠7 Inequality Theorem 2 (AaSs)11. C 3. Each of Xylie, Marie, Angel and Chloe was given an 18-inch piece of stick. • Considering the triangle with sides p, q and s: They were instructed to create a triangle. Each cut the stick in their own chosen The angle opposite p is 61°. Hence, s < q < p. lengths as follows: Xylie—6 in, 6 in, 6 in; Marie—4 in, 5 in, 9 in; Angle—7 in, 5 • Considering the triangle with sides r, s and t: in, 6 in; and Chloe—3 in, 7 in, 5 in. Who among them was not able to make a The angle opposite r is 60°. Hence, t < r < s. triangle? Combining both results, t < r < s < q < p Basis: Triangle Inequality Theorem (AaSs) a. Xylie b. Marie c. Angel d. Chloe 4. What are the possible values for x in the figure? 12 A 11 SF (4x–3)0 420 E a. x < 11.25 c. x ≤ 11.25 b. x > 11.25 d. x ≥ 11.25 410

12. C Basis: Triangle Inequality Theorem (SsAa) 5. From the inequalities in the triangles shown, a conclusion can be reached using13. C Basis: Triangle Inequality Theorem (AaSs) the converse of hinge theorem. Which of the following is the last statement?14. B Basis: Triangle Inequality Theorem 3 (S1 + S2 > S3) O15. B 10 816. D HM17. D 10 8 24 18 E18. C III mostly conveys wrong signal to a client. a. HM ≅ HM c. HO ≅ HE 19. A b. m∠OHM > m∠EHM d. m∠EHM > m∠OHM20. D 6. Hikers Oliver and Ruel who have uniform hiking speed walk in opposite directions- Oliver, eastward whereas Ruel, westward. After walking three kilometers each, both of them take left turns at different angles- Oliver at an angle of 300 and Ruel at 400. Both continue hiking and cover another four kilometers each before taking a rest. Which of the hikers is farther from their point of origin? a. Ruel c. It cannot be determined. b. Oliver d. Ruel is as far as Oliver from the rendezvous. 7. Which of the following is the accurate illustration of the problem? a. b. c. d. 411

Teacher’s Note and Reminders 8. The chairs of a swing ride are farthest from the base of the swing tower when the swing ride is at full speed. What conclusion can you make about the angles of the swings at different speeds? a. The angles of the swings remain constant whether the speed is low or full. b. The angles of the swings are smaller at full speed than at low speed. c. The angles of the swings are larger at full speed than at low speed. d. The angles of the swings are larger at low speed than at full speed. 9. Will you be able to conclude that EM > EF if one of the following statements is not established: AE ≅ AE, AF ≅ AM, m∠MAE > m∠FAE? a. Yes, I will. A M b. No, I won’t. 36o 42o c. It is impossible to decide. d. It depends on which statement is left out. E F 10. Which side of ∆GOD is the shortest? O 86o a. GO c. DG b. DO d. GD G 49o D 11. The diagram is not drawn to scale. Which of the following combined inequalities describes p,q,r,s, and t? Don’t a. p<q< r <s< t q tForget! s<p<q< r < t 59o 61o b. t <r <s<q<p q<p< t < r <s Ps c. 60o 59o d. r 412

Teacher’s Note and Reminders 12. In ∆TRU, TR = 8 cm, RU = 9 cm, and TU = 10 cm. List the angles in order from least to greatest measure. Don’t Forget! a. m∠T, m∠R, m∠U c. m∠R, m∠T, m∠U b. m∠U, m∠T, m∠R d. m∠U, m∠R, m∠T 13. List the sides of ∆LYK in order from least to greatest measure. K 84o Y 58o 38o L a. LY, YK, LK c. LY, LK, KL b. YK, YL, LK d. YK, LK, LY 14. What is the range of the values of the diagonal d of a lot shaped like a parallelogram if adjacent sides are 10 inches and 14 inches? a. 4 ≥ d ≥ 24 c. 4 ≤ d ≤ 24 d. 4 > d > 24 b. 4 < d < 24 For items no. 15-20, use the figure shown. 15. A balikbayan chose you to be one of the contractors to design an A-frame house maximizing the size of two square lots with dimensions 18 ft and 24 ft on each side. Which of the following is affected by the dimensions of the lot if the owner would like to spend the same amount of money on the roofs? I. The width of the base of the house frames II. Design of the windows III. The height of the houses IV. The roof angles a. I and IV c. II, III and IV b. III and IV d. I, II, III, and IV 413

Teacher’s Note and Reminders 16. Which of the following theorems justifies your response in item no. 15? Don’t I. Triangle Inequality Theorem 1 Forget! II. Triangle Inequality Theorem 2 III. Triangle Inequality Theorem 3 IV. Hinge Theorem V. Converse of Hinge Theorem a. I, II, and III b. IV only c. IV and V d. V only 17. If the owner would like the same height for both houses, which of the following is true? I. Roof costs for the larger lot is higher than that of the smaller lot. II. The roof of the smaller house is steeper than the larger house. a. I only c. neither I nor II b. II only d. I and II 18. What considerations should you emphasize in your design presentation so that the balikbayan would award you the contract to build the houses? I. Kinds of materials to use considering the climate in the area II. Height of floor-to-ceiling corner rooms and its occupants III. Extra budget needed for top-of-the-line furnishings IV. Architectural design that matches the available funds V. Length of time it takes to finish the project a. I, II, and IV c. I, II, IV, and V b. I, IV, and V d. I, II, III, IV, V 19. Why is it not practical to design a house using A-Frame style in the Philippines? I. A roof also serving as wall contributes to more heat in the house. II. Placement of the windows and doors requires careful thinking. III. Some rooms of the house would have unsafe low ceiling. IV. An A-Frame design is an unusually artful design. a. I and III c. I, II, and III b. II and IV d. I, II, III, IV 20. Why do you think an A-Frame House is practical in countries with four seasons? A. The design is customary. B. An artful house is a status symbol. C. The cost of building is reasonably low. D. The snow glides easily on steep roofs. 414

Before engaging the students in the different activities you have to underscore 1Lesson Inequalities inthe following to the students: TrianglesMathematical Connection — learning new lessons requires the use of lessons WWhhaatt ttoo KKnnoowwpreviously learned;Cooperative Learning — learning is much easier, faster, more meaningful and Let’s start the module by doing three activities that will reveal your backgroundmore fun when working with group mates; knowledge on triangle inequalities.Engagement — learning is maximized through active performance of studentsin all activities Activity 1 MY DECISIONS NOW AND THEN LATERActivity No.1: Directions: My Decisions Now and Then Later 1. Replicate the table below on a piece of paper.Let them perform activity No. 1 in at most 5 minutes. Inform the students that 2. Under the my-decision-now column of the first table, write A if you agree with thethere is no right or wrong answer because the activity is only intended to find statement and D if you don’t.out their background knowledge on Inequalities in Triangles. Tell them these:Your answers can be modified after tackling the module. Hence, there will be 3. After tackling the whole module, you will be responding to the same statementsno checking of your responses. Hence, the answer key that follows is used tocheck their final answers after tackling the module. using the second table. Answer Key to Activity No.1 Statement My Decision Now 1. D 2. D (can be should be replaced with is always) 1 To form a triangle, any lengths of the sides can be used. 3. D 4. A 2 The measure of the exterior angle of a triangle can be greater than the measure 5. A of its two remote interior angles. 3 Straws with lengths 3 inches, 4 inches and 8 inches can form a triangle. 4 Three segments can form a triangle if the length of the longest segment is greater than the difference but less than the sum of the two shorter segments. 5 If you want to find for the longest side of a triangle, look for the side opposite the largest angle. Statement My Decision Later 1 To form a triangle, any lengths of the sides can be used. 2 The measure of the exterior angle of a triangle can be greater than the measure of its two remote interior angles. 3 Straws with lengths 3 inches, 4 inches and 8 inches can form a triangle. 4 Three segments can form a triangle if the length of the longest segment is greater than the difference but less than the sum of the two shorter segments. 5 If you want to find for the longest side of a triangle, look for the side opposite the largest angle.415

Artistically Yours Activity 2 ARTISTICALLY YOURS! More TriangularYour task is to get students interested in the new lesson. You may start by Designs andposing this task: What objects around us are triangular in shape? You and yourstudents will find out that most objects are circular or rectangular. Artworks 1. Triangular Girl byAfter a 2-minute discussion, divide the class into groups and let them study thepictures and answer the questions in ponder time of Activity No. 2 Artistically Caroline JohanssonYours for at least three minutes. Let all group representatives report theiranswers to the questions. Give each representative at most one minute each Direction: Study the artworks below and answer the questions that h t t p : / /to be able to maximize time. Process all their answers by unifying all their follow:ideas or supplementing them so it would converge to the expected answers thecarolinejohansson.provided. c o m / b l o g / 2 0 11 / 1 0 /Invite also the students to discover more triangular designs and artworks bylocating them in www.google.com under Images. Instruct them to type any triangular-girl-2/of the following in the search bar: triangular designs, triangular artworks,triangular architecture, triangular art, and more. 2. Tile works: Answer Key to Activity No.2 Diminishing Triangles1. Triangles http://sitteninthehills64.2. Yes. Some sides are longer than the others and some corners are larger blogspot.com/2010/05/ than the others.3. Possible Answers: Interesting, Practical, creative, artful tile-house-8.html4. Because they have not tackled the lesson yet, possible Answer: 3. Repetitive Graduation Inequalities in triangles in these artworks and designs are necessary in order to achieve beauty, artistry, creativity, and usefulness to the designs. by Scott Mihalik Teacher’s Note and Reminders http://mathtourist. Don’t blogspot.com/2012/02/ Forget! flight-of-tetrahedra.html 4. Maths-the best use for golf balls http://www.whizz.com/ blog/fun/maths-best- use-for-golf-balls/ 5. Luxury sailboat http://edgeretreats. com/ 6. Triangle Card Stand http://www.behance. net/gallery/ TRIANGLE-CARD- STAND/3883741 7. Triangular Periodic Table http://www.meta- synthesis.com/ webbook/35_pt/ pt_database.php?PT_ id=40 8. A triangular approach to fat loss by Stephen ?E S T I O Tongue http://www.flickr.com/ 1. What features prevail in the artworks, tools, photos/32462223@ equipment, and furniture shown? QU NS N05/3413593357/in/ 2. Have you observed inequalities in triangles in the designs? Explain. photostream/ 3. What is the significance of their triangular 9. Triangular Petal Card designs? http://www.flickr.com/ 4. How can you justify inequalities in triangles in these designs? photos/32462223@ N05/3413593357/in/ photostream/ Activity 3 HELLO, DEAR CONCEPT CONTRACTOR! Question: • Which among these The figure on the next page is a concept museum of inequalities in triangles. You will be constructing this concept museum throughout this designs and artworks module. you find most interesting? Explain. Each portion of the concept museum, mostly triangular, poses a • Which design you task for you to perform. All tasks are related to knowledge and skills you would like to pattern should learn about inequalities in triangles. from for a personal project? What is a contractor? A contractor is someone who enters into a binding agreement to build things. ~Wordweb 4.5a by Anthony Lewis~ What is a museum? Museum is a depository for collecting and displaying objects having scientific or historical or artistic value. ~Wordweb 4.5a by Anthony Lewis~416

Activity No.3: Note that the triangles in this concept museum are not drawn to scale and all sidesHello, Dear Concept Contractor! can be named using their endpoints. Consider using numbers to name the angles of these triangles.Your task is to make students understand the activity. To do that, these are thethings that you need to do: Notice that markings are shown to show which angles are larger and which sides are longer. These markings serve as your hints and clues. Your responses to the tasks must• Check and strengthen their understanding of the definitions of contractor be justified by naming all the theorems that helped you decide what to do. and museum; How many tasks of the concept museum can you tackle now?• Explain that the finished concept museum will display all the concepts and skills about inequalities in triangles and seeing the tasks at this point TH E provides them an overview of the lesson. Write two Knowing TH>TX>HX, what Write three inequalities to Write two• Point out that building the concept museum takes time—that there’s a Inequalities to question involving describe the sides of Inequalities to possibility that they may not be able to do any of the tasks listed on the describe angle 1. this triangle describe angle 2. triangles yet but they know already what to expect to learn. Thus, at the end inequality should you of the lesson, they will be able to encapsulate all the concepts and skills on use to check if inequalities in triangles using the concept museum. they form a• Let them see that in order for them to completely build the concept museum, triangle? MY they need to perform all the activities in the succeeding sections. 1 X CONCEPT N 2You need to master the concepts and skills of the whole module. To facilitate M 4Cthat, study the completely built concept museum. Note that the students must Write the combined TRIANGMLUESIENUEMQUonALITIES3 Write an if-thenhave built their concept museums at the end of this lesson. inequality you will statement about use to determine Come visit now! the sides given theIn short, the students have the option not to perform any task yet. The activityis just for presentation in order to direct the students of one goal—to build the the length of marked anglesconcept museum as the lesson unfolds. MK?The presentation of this activity must take at least five minutes. K Write if-then B6 Write if-then 5 Teacher’s Note and Reminders 7R Write a detailed if- tshtaethteaemnsmigdeleaensrstk.agebidvoeuntabosautnatWttiehfr-meittheeaennntglesstthatehteeasmindmgeealnsertskga.eibvdoeunt Don’t then statement to Write a detailed Forget! describe triangles given the marked if-then statement to MXK and KBF if describe triangles angle X is larger sides. than angle B MXK and KBF if MK is longer than KF. FW Replicate two (2) copies of the unfilled concept museum. Use the first one for your responses to the tasks and the second one for your justifications. TH E . MY CONCEPT 1 X MUSEUM N 2 M on TRIANGLE 3 4C INEQUALITIES Come visit now! 5 K B6 7R FW417

A. Responses Answer Key to Activity No.3 Are you excited to completely build your concept museum, Dear Concept Contractor? The only way to do that is by doing all the succeeding activities in the TH E next section of this module. The next section will also help you answer this essential question raised in the activity Artistically Yours: How can you justify inequalities in∠1 > ∠MTX Is TH > TX > HX > TH? EH + EN > HN ∠2 > ∠CEN triangles?∠1 > ∠MXT EH + HN > EN ∠2 > ∠CNE The next lesson will also enable you to do the final project that is inspired by the EN + HN > EH artworks shown in Artistically Yours. When you have already learned all the concepts and skills related to inequalities in triangles, you will be required to make a model of MY a folding ladder and justify the triangular features of its design. Your design and its justification will be rated according to these rubrics: accuracy, creativity, efficiency, and 1 X CONCEPT N 2 mathematical justification.M 4C TRIANGMLUESIENUEMQUonALITIES3 WWhhaatt ttoo PPrroocceessss KX − MX < MK < KX If ∠5 > ∠4 > ∠3, Come visit now! Your first goal in this section is to develop and verify the theorems on inequalities in + MX then CN > NR > CR triangles. To succeed, you need to perform all the activities that require investigation. K If BF > BK, then B6 If ∠7 > ∠6, then 5 When you make mathematical generalizations from your observations, you are ∠BKF > ∠BF BW > RW 7R actually making conjectures just like what mathematicians do. Hence, consider yourself If <X of ∆MXK > If little mathematicians as you perform the activities. ∠B of ∆KBF, then BW If MK of ∆MXK > MK > FK FK of ∆KBF, then Once you have developed these theorems, your third goal is to prove these > BF > FW, ∠MXK > ∠KBF theorems. You have to provide statements and/or reasons behind statements used to then ∠BFW > deductively prove the theorems. ∠BWF > ∠FBW. The competence you gain in writing proofs enables you to justify inequalities in FW triangles and in triangular features evident in the things around us.B. Justifications Before you go through the process, take a few minutes to review and master again the knowledge and skills learned in previous geometry lessons. The concepts and TH E skills on the following topics will help you succeed in the investigatory and proof-writing activities. Exterior Angle Triangle Inequality Triangle Inequality Exterior AngleInequality Theorem Theorem 3 Theorem 3 Inequality Theorem 1. Axioms of Equality 1.1 Reflexive Property of Equality (S1 + S2 > S3) (S1 + S2 > S3) • For all real numbers p, p = p. MY 1.2 Symmetric Property of Equality 1 X CONCEPT N 2 • For all real numbers p and q, if p = q, then q = p.M 4C 1.3 Transitive Property of Equality TRIANGMLUESIENUEMQUonALITIES3 Triangle Inequality Triangle Inequality • For all real numbers p, q, and r, if p = q and q = r, then p = r. Come visit now! 1.4 Substitution Property of Equality Theorem 3 Theorem 2 • For all real numbers p and q, if p = q, then q can be substituted for p in any (S1 + S2 > S3) (AaSs) expression. K Triangle Inequality B6 5 418 Hinge Theorem 7R Theorem 1 Triangle Inequality (SsAa) Triangle Theorem 2 Converse of Hinge (AaSs) Theorem Inequality Theorem 1 (SsAa) FW

WHAT TO PROCESS: 2. Properties of Equality 2.1 Addition Property of EqualityThe PROCESS section showcases investigatory activities designed to developand verify the theorems to learn in the lesson. This is also where students are • For all real numbers p, q, and r, if p = q, then p + r = q + r.given the opportunities to practice the concepts and skills learned in the lesson 2.2 Multiplication Property of Equalityand to write proofs of the theorems. This section is characterized by student-centered activities as inspired by this 3. • For all real numbers p, q, and r, if p = q, then pr = qr.saying of Kahlil Gibran: Definitions, Postulates and Theorems on Points, Lines, Angles and Angle PairsThe teacher who is indeed wise does not bid you to enter the house of his wisdom 3.1 Definition of a Midpointbut rather leads you to the threshold of your mind. • If points P, Q, and R are collinear (P–Q–R) and Q is the midpoint of PR, thenYour task in this section is to make sure that all the group activities that are PQ ≅ QR.suggested in the learning module shall be completely delivered. 3.2 Definition of an Angle BisectorYour responsibilities involve the following: • If QS bisects ∠PQR, then ∠PQS ≅ ∠SQR.1. Conduct a quick but comprehensive review of the pre-requisite skills 3.3 Segment Addition Postulate needed to succeed in the new lesson; and • If points P, Q, and R are collinear (P–Q–R) and Q is between points P and R, then PQ + QR ≅ PR.2. Manage group work. 2.1 Grouping of students 3.4 Angle Addition Postulate Suggestion: Form at least two sets of groupings so students will learn to • If point S lies in the interior of ∠PQR, then ∠PQS + ∠SQR ≅ ∠PQR. work with different group mates 2.2 Time allotment for each group work 4. 3.5 Definition of Supplementary Angles Note: Activities in this section are simplified so that they can be performed • Two angles are supplementary if the sum of their measures is 180º. in a short span of time. It can be done individually or as a group. Suggestion: For the students to finish the whole module on time (within 3.6 Definition of Complementary Angles two weeks), you may opt to let groups do the activity in their free time or • Two angles are complementary if the sum of their measures is 90º. after class. Let them write their answers to questions in Ponder Time on a piece of manila paper. 3.7 Definition of Linear Pair • Linear pair is a pair of adjacent angles formed by two intersecting lines 2.3 Do’s and don’ts during group work 2.4 Monitor student behavior during group work to ensure time is spent 3.8 Linear Pair Theorem on the task. • If two angles form a linear pair, then they are supplementary. 2.5 Give technical assistance during group work so that group responses to activities and process questions are accurate. In 3.9 Definition of Vertical Angles short provide subtle coaching. • Vertical angles refer to two non-adjacent angles formed by two intersecting Suggestion: If you let the groups answer the activity as an assignment, lines give them at least 3-5 minutes to review their answers so you may be able to give them technical assistance if their outputs have errors. 3.10 Vertical Angles Theorem • Vertical angles are congruent. How to Measure Angles using a Protractor 9070 80 100 110 120 Internet Learning 100 80 60 130 60 110 70 Mastering the Skill in Estimating Measures of Angles 50 120 40 50 160 Interactive: 130 150 20 • http://www.mathplayground.com/measuringangles.html 140 30 • http://www.teacherled.com/resources/anglemeasure/ 140 40 anglemeasureload.html 30 Games: 150 • http://www.bbc.co.uk/schools/teachers/ks2_activities/ 20 160 maths/angles.shtml • http://www.innovationslearning.co.uk/subjects/maths/ 10 170 170 10 activities/year6/angles/game.asp • http://www.bbc.co.uk/keyskills/flash/kfa/kfa.shtml 0 180 • http://resources.oswego.org/games/bananahunt/bhunt.html 180 0 • http://www.fruitpicker.co.uk/activity/ Origin Base Line419

2.6 Processing of outputs in group work To measure an angle, the protractor’s origin is placed over the Mathematical Suggestion: Let them post their work for everyone to see. If groups have vertex of an angle and the base line along the left or right side of the similar answers, you may decide (or let the class decide) only one or two angle. The illustrations below show how the angles of a triangle are History groups to discuss their answers to questions in Ponder Time. If there are measured using a protractor. Who invented the groups with different answers, let the class discuss these answers. Note that a good teacher facilitator minimizes unexpected answers by giving first advanced technical assistance to every group before posting outputs. protractor?Your facilitating role is crucial so that students are able to achieve the goal in this 60 9070 80 100 110 60 9070 80 100 110section to develop, verify, and prove all six theorems of inequalities in triangles 50 120 100 80 70 120 50 120 100 80 70 120and to continue to unlock triangles in their concept museum. 110 110 130 60 130 130 60 130 50 50 40 160 40 160 150 20 150 20 Capt. Joseph Huddart 140 140 30 140 140 30 (1741-1816) of the United 40 40 States Navy invented the 30 30 first advanced protractor 150 150 in 1801. It was a three- arm protractor and was 20 160 20 160 used for navigating and determining the location 10 32o 170 10 40o 170 of a ship 170 10 170 10 0 180 0 180 180 0 180 0 0 180 170 10 20 ~Brian Brown of www. 160 ehow.com~ 108o 30 150 40 140WWhhaatt ttoo PPrroocceessss 130 50 To read more about the history of protractor, visit Your first goal in this section is to develop and verify the theorems on 180 60 inequalities in triangles. To succeed, you need to perform all the activities that 170 10 0 120 these website links: require investigation. 110 When you make mathematical generalizations from your observations, you are actually making conjectures just like what mathematicians do. Hence, 80 100 70 consider yourself little mathematicians as you perform the activities. • http://www.counton.1602090 Once you have developed these theorems, your third goal is to prove these org/museum/floor2/150140110100 theorems. You have to provide statements and/or reasons behind statements gallery5/gal3p8.html30401301207080 used to deductively prove the theorems. 50 60 • h t t p : / / w w w . The competence you gain in writing proofs enables you to justify a b l o g a b o u t h i s t o r y. inequalities in triangles and in triangular features evident in the things around c o m / 2 0 11 / 0 7 / 2 9 / t h e - us. worlds-first-protractor/ Before you go through the process, take a few minutes to review and 5. Definitions and Theorems on Triangles master again the knowledge and skills learned in previous geometry lessons. 5.1 The sum of the measures of the angles of a triangle is 180º. The concepts and skills on the following topics will help you succeed in the 5.2 Definition of Equilateral Triangle investigatory and proof-writing • An equilateral triangle has three sides congruent. 5.3 Definition of Isosceles Triangle • An isosceles triangle has two congruent sides. • Is an equilateral triangle isosceles? Yes, since it also has two congruent sides. • Base angles of isosceles triangles are congruent. • Legs of isosceles triangles are congruent. 5.4 Exterior Angle of a Triangle • An exterior angle of a triangle is an angle that forms a linear pair with an interior angle of a triangle when a side of the triangle is extended. 5.5 Exterior Angle Theorem • The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles of the triangle. 5.6 Sides and Angles of a Triangle • ∠S is opposite EC and EC is opposite ∠S. • ∠E is opposite SC and SC is opposite ∠E • ∠C is opposite ES and ES is opposite ∠C. 420

Notes to the Teacher Internet Learning 6. Definition and Postulates on Triangle Congruence Mastering the For the review of all the pre-requisite concepts of the lesson on inequality of 6.1 Definition of Congruent Triangles: Corresponding parts oftriangles, you may decide to present it in a creative manner like making sets of Triangle Congruenceflashcards for each of the following: Postulates congruent triangles are congruent (CPCTC).Set 1: Axioms of EqualitySet 2: Properties of Equality Video 6.2 Included AngleSet 3: Definitions, Postulates, and Theorems on Points, Lines, Angles, and • http://www.onlinemathlearn- • Included angle is the angle formed by two distinct sides of Angle Pairs ing.com/geometry-congru-Set 4: Definitions and Theorems on Triangles ent-triangles.html a triangle.Set 5: Definitions and Postulates on Triangle Congruence InteractiveSet 6: Properties of Inequality • http://www.mrperezonlin Y emathtutor.com/G/1_5_Prov-Sample Flash Card (Front and Back): ing_Congruent_SSS_SAS_ • ∠YES is the included angle of EY and ES ASA_AAS.html • ∠EYS is the included angle of YE and YS E • http://nlvm.usu.edu/ • ∠S is the included angle of SE and SY en/nav/frames_ asid_165_g_1_t_3. html?open=instructions • http://www.mangahigh.com/ en/maths_games/shape/ congruence/congruent_ triangles?localeset=en S 6.3 Included Side W E • Included side is the side common to two angles of a triangle. • AW is the included side of ∠WAE and ∠EWA • EW is the included side of ∠AEW and ∠AWE A• If points P, Q and R are • AE is the included side of ∠WAE and ∠AEW collinear and Q is the midpoint of PR, then 6.4 SSS Triangle Congruence Postulate PQ ≅ QR Definition of a Midpoint 6.5 SAS Triangle Congruence Postulate 6.6 ASA Triangle Congruence Postulate P QR 7. Properties of Inequality 7.1 For all real numbers p and q where p > 0, q > 0:You may also assign each group of students to prepare a specific set of flashcards using used folders. You may then have a quiz bee for six representatives • If p > q, then q < p.of a group using the flash cards. If a competitor is the first one to name the • If p < q, then q > p.axiom, property, definition, theorem or postulate flashed, he then can make a 7.2 For all real numbers p, q, r and s, if p > q and r ≥ s, then p + r > q + s.step forward until he/she who reaches the front of the classroom is declared as 7.3 For all real numbers p, q and r, if p > q and r > 0, then pr > qr.winner. Another set of representatives is called until everyone has mastered all 7.4 For all real numbers p, q and r, if p > q and q > r, then p > r.the axioms, properties, definitions, theorems, and postulates. 7.5 For all real numbers p, q and r, if p = q + r, and r > 0, then p > q.Include in your discussion Capt. Joseph Huddard—the inventor of the firstadvanced protractor. In that connection, invite them to visit presented website The last property of inequality is used in geometry such as follows:links about protractors and those that have interactive activities and games thatenable them to master the skill in estimating measures of angles and knowledge P QR P 12of triangle congruence postulates. In this manner, their internet visits would bemore educational. Follow up on their Internet activity by asking them to share Q is between P and R. QRtheir insights about learning mathematics online. PR ≅ PQ + QR ∠1 and ∠2 are adjacent angles. Then PR > PQ and PR > QR. ∠PQR ≅ ∠1 + ∠2 Then ∠PQR > ∠1 and ∠PQR > ∠2 421

Teacher’s Note and Reminders 8. How to Combine Inequalities • Example: How do you write x < 5 and x > -3 as a combined inequality? Don’t Forget! x > -3 x<5 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 From the number line, we observe that the value of x must be a value between -3 and 5, that is, x is greater than -3 but less than 5. In symbols, -3 < x < 5. 9. Equality and Congruence Congruent figures (segments and angles) have equal measures such that: • If PR ≅ PR, then PR = PR. • If ∠PQS ≅ ∠PQS, then m∠PQS = m∠PQS. Note that to make proofs brief and concise, we may opt to use PR ≅ PR or ∠PQS ≅ ∠PQS instead of PR = PR or m∠PQS = m∠PQS. Because the relation symbol used is for congruence; instead of writing, say, reflexive property of equality as reason; we just have to write, reflexive property. Note that some other books sometimes call reflexive property as reflexivity. 10. How to Write Proofs Proofs in geometry can be written in paragraph or two-column form. A proof in paragraph form is only a two-column proof written in sentences. Some steps can be left out when paragraph form is used so that two-column form is more detailed. A combination of both can also be used in proofs. The first part can be in paragraph form especially when the plan for proof is to add some constructions first in the illustration. Proving theorems sometimes requires constructions to be made. The first column of a two-column proof is where you write down systematically every step you go through to get to the conclusion in the form of a statement. The corresponding reason behind each step is written on the second column. Possible reasons are as follows: Given, by construction, axioms of equality, properties of equality, properties of inequality, definitions, postulates or previously proven theorems. 422

Activity No.4: The following steps have to be observed in writing proofs:What if It’s Longer? • Draw the figure described in the problem. The figure may already be drawn forFor Activity No. 4, make sure that each student has his/her own you, or you may have to draw it yourself.protractor. Ask them to define precision, accuracy, and toleranceusing their own words. Discuss the meaning of these words • Label your drawn figure with the information from the given byrelated to making measurements by giving Donna Roberts’sdefinitions:  marking congruent or unequal angles or sides,The precision of a measuring instrument is determined by the  marking perpendicular, parallel or intersecting lines orsmallest unit to which it can measure. The precision is said to  indicating measures of angles and/or sidesbe the same as the smallest fractional or decimal division on thescale of the measuring instrument. The markings and the measures guide you on how to proceed with the proof andAsk the students: What is the precise unit of a ruler? Answer it also directs you whether your plan for proof requires you to make additional con-should be millimeter. structions in the figure.Accuracy is a measure of how close the result of themeasurement comes to the “true”, “actual”, or “accepted” value. • Write down the steps carefully, without skipping even the simplest one. Some ofAccuracy answers this question: How close is your answer to the first steps are often the given statements (but not always), and the last stepthe accepted value? is the statement that you set out to prove.Tolerance is the greatest range of variations in measurementsthat can be allowed. 11. How to Write an Indirect ProofTolerance addresses this question: How much error in the 11.1 Assume that the statement to be proven is not true by negating it.answer is acceptable? 11.2 Reason out logically until you reach a contradiction of a known fact.Proceed by discussing that it is expected that the measurements 11.3 Point out that your assumption must be false, thus, the statement to be proventhey get from measuring the same lengths vary. Explain that must be true.their answers are not wrong. Their answers vary becausea measurement made with a measuring device is 12. Greatest Possible Error and Tolerance Interval in Measurementsapproximate, not exact. Discussion of the Greatest Possible You may be surprised why two people measuring the same angle or length may giveError and Tolerance Interval should follow. different measurements. Variations in measurements happen because measurementNote that you and your class may decide on a tolerance interval. with a measuring device, according to Donna Roberts (2012), is approximate. ThisFor the example given in the learning guide, you may decide to variation is called uncertainty or error in measurement, but not a mistake. She added that there are ways of expressing error of measurement. Two are the following: Greatest Possible Error (GPE) One half of the measuring unit used is the greatest possible error. For example, you measure a length to be 5.3 cm. This measurement is to the nearest tenth. Hence, the GPE should be one half of 0.1 which is equal to 0.05. This means that your measurement may have an error of 0.05 cm, that is, it could be 0.05 longer or shorter. Tolerance Intervals Tolerance interval (margin of error) may represent error in measurement. This interval is a range of measurements that will be tolerated or accepted before they are considered flawed. Supposing that a teacher measures a certain angle x as 36 degrees. The measurement is to the nearest degree, that is, 1. The GPE is one half of 1, that is, 0.5. Your answer should be within this range: 36-0.5 ≤ x ≤ 36 + 0.5. Therefore, the tolerance interval or margin of error is 35.5≤x≤36.5 or 35.5 to 36.5. 423

have this margin of error: 36 − 1 ≤ x ≤ 36 + 1 or 35 ≤ x ≤37. Thus, Now that you have already reviewed concepts and skills previously learned that area student’s measure maybe 35 or 37 degrees. Still the answer is useful in this module, let us proceed to the main focus of this section—develop, verify,accepted because you set which range of measurements has to and prove the theorems on inequalities in triangles.be tolerated.In the discussion of errors in measurement, let the groups do the Activity 4 WHAT IF IT’S LONGER?activity. Let them post their outputs. Process them and answersto numbers 3, 4, and 5 should be written on cartolina and posted Materials Needed: protractor, manila paper, ruleron the display board for math concepts developed. Procedures: 1. Replicate the activity table on a piece of manila paper.Once all questions are answered, let the students answer Quiz 2. Measure using a protractor the angles opposite the sides with given lengths. IndicateNo. 1. Be sure to explain fully and carefully answers to eachitem in order to strengthen their understanding of the topic. For the measure in your table.the questions under each item in Enrichment, you may let them 3. Discover the relationship that exists between the lengths of the sides of triangles andanswer by group. Give it as an assignment to give the studentstime to think. Follow this procedure in unlocking the answers to the angles opposite them and write them on your piece of manila paper.all quizzes in this module. F T Answer Key to Activity No.4 5 10 1. Yes, there is. 6P 2. When one side of a triangle is longer than a second side, the angle 3.5 opposite the first side is larger than the angle opposite the second side. U 4.5 N Y R5 Y 3. If one side of a triangle is longer than a second side, then the angle Triangle Length of Sides Measures of Angles opposite the first side is larger than the angle opposite the second side. ∆FUN FN 3.5 Opposite the Sides 4. If one side of a triangle is the longest, then the angle opposite it is the NU 4.5 m∠U ∆PTY TP 5 largest. PY 6 m∠F 5. If one side of a triangle is the shortest, then the angle opposite it is the m∠Y smallest. 6. Note: Because of GPE and Tolerance Interval, it is your task to give the m∠T measure of the sides as accurately as you can. ∆RYT RY 5 m∠T TY 10 m∠R424

Mathematics in the Kitchen: The Kitchen Triangle Mathematics in the ?E S T I O 1. Is there a relationship between the length of a side1. Kitchen QU NS of a triangle and the measure of the angle opposite The Kitchen Triangle it? Yes, there is. No, there isn’t. SR = SC < RC RC > RS > SC RC > RS > SC RS > RC > SC The Kitchen Triangle was 2. Making Conjecture: What is the relationship ∠C = ∠C < S ∠S > ∠C > R ∠S > ∠C > R ∠C > ∠S > R developed in the 1950s between the sides of a triangle and the angles as a tool to aid designers opposite them?2. Possible answer and reasons: in creating an effective • When one side of a triangle is longer than kitchen layout. The triangle a second side, the angle opposite the 1. It utilizes only two corners of a has a corner at the sink, _________. room. the refrigerator, and the stove, the three essential 3. Your findings in no. 2 describe the Triangle 2. It is more open so two or more locations in the kitchen when Inequality Theorem 1. Write it in if-then form. people can watch and learn and cooking. Most kitchen plans • If one side of a triangle is also help in cooking still include this today. The longer than a second side, then idea is to have them close ____________________________. Answer Key to Quiz No. 1 enough that they can easily be moved between, but not 4. What is the relationship between the longest sideA. too far away to reduce the of a triangle and the measure of the angle opposite amount of movement while it? cooking. The general rule is that the triangles perimeter 5. What is the relationship between the shortest side must be at least 12 ft., but of a triangle and the measure of the angle opposite should be larger than 26 ft. it? The area inside the triangle 6. Without using a protractor, determine the measure should be completely open, of the third angles of the triangles in this activity. making movement between each of these easy. Most (Hint: The sum of the measures of the angles of a modern kitchen plans still triangle is 180°.) include this. Triangle Largest Angle Smallest Angle Name of Working Measure of the Third See the whole article at Triangle Equations Angle http://bathroomphotogallery. 1. ∆AIM ∠A ∠I com/kitchen-plans.php~ ∆FUN m∠N ∠R 2. ∆RYT ∠Y ∠D Questions: ∆TYP m∠P 1. Suppose R is for 3. ∆END ∠N ∆WHT ∆TRY m∠Y HW, HT, TW Refrigerator, S is for Sink,B. ∠T, ∠W, ∠H and C for cooking stove, describe the relationship of the sides and angles of the kitchen triangle in each of the model. 2. If you were to build a house in the future, which kitchen model do you prefer from the figure? Why? Quiz No. 1 ∆NAY ∆FUN Directions: Write your answer on a separate answer sheet. Sides NY, AN, AY FN, FU, NU A. Name the smallest angle and the largest angle of the following triangles: Angle ∠A, ∠Y, ∠N ∠U, ∠N, ∠FC. Triangle Largest Smallest Angle Angle Grant: Grant: 1. ∆AIM 3 wishes 2 wishes 2. ∆END 3. ∆RYT Region to Hit with Region O Region M an Arrow 425

Mathematics in Art: Geometric Shapes for Foundation B. The diagrams in the exercises are not drawn to scale. If each diagram were drawn Piecing to scale, list down the sides and the angles in order from the least to the greatest measure.1. Possible Answer: The figures started from the largest regular polygons. ∆NAY ∆FUN ∆WHT 1.1 Hexagon 1.1.1 The next larger hexagon is determined by doing the following: Sides Angle • getting the midpoints of the sides of the original hexagon • connecting adjacent midpoints to form segments that serve as C. Your parents support you in your studies. One day, they find out that your topic in Grade 8 Math is on Inequalities in Triangles. To assist you, they attach a sides of the next hexagon. triangular dart board on the wall with lengths of the sides given. 1.1.2 Repeat steps in 1.1 until the desired smallest hexagon is formed They say they will grant you three wishes if you can hit with an arrow the corner 1.2 Heptagon with the smallest region and two wishes if you can hit the corner with the largest 1.2.1 The next larger heptagon is determined by doing the following: region. • placing a point of desired distance from the left endpoint of the • Which region should you hit so your parents will grant you three wishes? sides of the original heptagon • Which region should you hit so your parents will grant you two wishes? • connecting adjacent points to form segments that serve as Mathematics in Art sides of the next heptagon. Geometric Shapes for 1.2.2 Repeat steps in 1.1 until the desired smallest heptagon is Foundation Piecing by formed Dianna Jesse2. Possible Answer: Grant: Grant: Challenge: Teacher’s Note and Reminders 3 wishes 2 wishes 1. Which figure is drawn Don’t Region to Hit with first in the artworks--the Forget! an Arrow smallest polygon or the largest polygon? 2. Make your own design by changing the positions or the lengths of the sides of the triangles involved in constructing the figure. 3. Would you like to try using the hexagon? Visit this web link to see the artworks shown: http:// diannajessie.wordpress. com/tag/triangular-design/~ 426

Activity No.5: Activity 5 WHAT IF IT’S LARGER?What if It’s Larger?For Activity No. 5, start up the class by having a review of the different kinds of Materials Needed: ruler, manila papertriangles according to sides and angles. Proceed by asking what triangles are Procedures:shown in the activity What if it’s Larger. 1. Replicate the activity table on a piece of manila paper.Discuss the GPE and the tolerance interval of measurements. Once these 2. Measure using ruler the sides opposite the angles with given sizes. Indicate theare established, let the groups proceed with the activity. Let them post theiroutputs, process their outputs; and their answers to the questions in ponder lengths (in mm) on your table.time. The answers to numbers 3, 4, and 5 should be written on cartolina and 3. Develop the relationship of angles of a triangle and the lengths of the sidesposted in a display board for math concepts. opposite them by answering the ponder questions on a piece of manila paper.After discussion, let them answer Quiz No. 1. L QO 54o 81o 48oAnswer Key to Activity No.5 Y 36o 90o F T 38o 61o U 103o 29o M G1. Yes, there is. Lengths of Sides Opposite2. When one angle of a triangle is larger than a second angle, the side Triangle Measure of the Angles the Angles ∆LYF opposite the first angle is longer than the side opposite the second angle. ∆QUT m∠L FY ∆OMG m∠Y LF3. If one angle of a triangle is larger than a second angle, then the side m∠F LY m∠Q TU opposite the first angle is longer than the side opposite the second angle. m∠U QT m∠T QU4. If one angle of a triangle is the largest, then the side opposite it is the m∠O MG m∠M GO longest. m∠G MO5. If one angle of a triangle is the smallest, then the side opposite it is the shortest.6. Name of Smallest Smaller Largest ?E S T I O Triangle Angle Angle Angle ∆LYF ∠Y ∠L ∠F ∆QUT ∠T ∠U ∠Q 1. Is there a relationship between the size of an angle and the length ∆OMG ∠M ∠O ∠G QU NS of the side opposite it? Yes, there is. No, there isn’t.7. 2. Making Conjecture: What is the relationship between the angles of Name of Shortest Shorter Longest a triangle and the sides opposite them? Triangle Side Side Side • When one angle of a triangle is larger than a second angle, ∆LYF LY FY LY the side opposite the _______________________________. ∆QUT TU QT QU 3. Your findings in no. 2 describe Triangle Inequality Theorem 2. Write it in if-then form. ∆OMG MO GM GO 4. What is the relationship between the largest angle of a triangle and the side opposite it? 5. What is the relationship between the smallest angle of a triangle and the side opposite it? 427

8. 6. Arrange in increasing order the angles of the triangles Mathematics for in this activity according to measurement. Kind of How do you know that a certain side is the longest side? Eco-Architecture Triangle Triangular Skyscraper Name of Smallest Smaller Largest with Vegetated Mini- Acute ∆ The longest side is opposite the largest acute angle. Triangle Angle Angle Angle Atriums Right ∆ The longest side is opposite the right angle. ∆LYF Obtuse ∆ The longest side is opposite the obtuse angle. ∆QUT ∆OMGAnswer Key to Quiz No. 2 Longest Side Shortest Side 7. Arrange in decreasing order the sides of the triangles The triangular form, which TY RY in this activity according to their lengths. in China is symbolic withA. AT AP balance and stability, also Triangle LV LU Name of Shortest Shorter Longest allows the building to shade Triangle Side Side Side itself, which lowers the 1. ∆TRY ∆FRE amount of energy required 2. ∆APT ∆LYF to cool the interiors. The 3. ∆LUV EF, FR , ER signature feature of the ∆QUT entire design is the atrium, which runs the entire height ∆OMG of the building and also allows each level to beB. ∆TRP ∆ZIP 8. Having learned Triangle Inequality 2, answer the illuminated by natural light. PT, PR and RT PZ, IZ, IP question in the table. List of Questions: Sides in Kind of How do you know that a certain side is Decreasing Triangle the longest side? 1. Have you seen triangular Order of buildings or structures in Lengths Acute ∆ your area? Right ∆ 2. When do you think it is best to use a triangularC. NZ Obtuse ∆ design like the oneD. Answers vary shown in building a structure? To find out the reasons why the triangular design is eco- friendly, visit this website link: http://www.ecofriend. com/eco-architecture- triangular-skyscraper- designed-with-vegetated- mini-atriums.html Mathematics in Eco-Architecture: Triangular Skyscraper QUIZ No. 21. Answers vary Directions: Write your answer on a separate answer sheet. Note that the diagrams in the2. Possible Answers: exercises are not drawn to scale. • When the lot to build on is triangular in shape. • When the owner would like to have a triangular design. A. Name the shortest side and the longest side of the following triangles: Triangle Longest Side Shortest Side ∆TRY 1. ∆APT 2. ∆LUV 3. 428

Triangular Design and Artworks Mathematics in the B. List down the sides in order from the longest to the shortest Garden length. 1. Answers Vary 2. Possible Answers: How to Space ∆TRP ∆ZIP ∆FRE • Triangular Petal Card because it is easy to perform Sprinklers? • Triangular Card Stand for those who likes wood working C. Skye buys a triangular scarf with angle measures as described • Triangular Girl for those who love sketching and drawing ~irrigationrepair.com~ in the figure shown. She wishes to put a lace around the • Diminishing Triangles for those who love tiling works. edges. Which edge requires the longest length of lace? A. Square SpacingActivity No.6:When Can You Say “ENOUGH”? Square spacing is the easiest to plot; the downfallTwo days before Activity No. 6 will be tackled, assign the groups to prepare is that there will be areaspieces of straws with the following lengths in inches: 3, 4, 5, 6, 7, 8, 9, 10, 11, that are going to be coveredand 12. Let them duplicate the 3- and 5-inch straws. Note: If straws are not by all four sprinkler headsavailable, they may use other objects like broom sticks. Make sure that all causing some over watering.groups have the materials on the day the activity is scheduled. Sprinklers are spaced relatively close when using a square pattern (on average around 50% of the diameter of the throw). This means you will also need more sprinkler heads to cover a given area. B. Triangular SpacingLet the group do the activity and record their findings in the table they transferred Triangular spacing is plottedon the manila paper. Process their outputs and their answers to the questions.Answers should be written on cartolina and posted on the display board of using three points whichmath concepts. means that more surfaceEnd the activity by asking this: What insight can you share about the title of theactivity? After sharing insights, let them answer Quiz No. 3. Activityarea is watered with less 6overlap .Since you can Teacher’s Note and Reminders WHEN CAN YOU SAY “ENOUGH!”?cover more surface using Don’t Forget! triangular spacing you will be able to space the sprinkler heads farther apart (usually around 60% of Materials Needed: plastic straws, scissors, manila paper, and ruler the diameter of the throw). Procedure:Using a triangular pattern in plotting sprinkler heads can save money because less sprinkler heads are needed 1. Cut pieces of straws with the indicated measures in inches. to irrigate any given area. Task: There are three pieces in each set. If you were asked to space 2. Replicate the table in this activity on a piece of manila paper. sprinklers, which spacing 3. With each set of straws, try to form triangle LMN. would you use? 4. Write your findings on your table and your responses to the To read more about ponder questions on a piece of manila paper. spacing sprinklers, visit this website link: http://www. L irrigationrepair.com/how_to_ space_rotors_sprays.html mn Nl M429

Answer Key to Activity No.6 Compare the sum of theSets of Straw Do the Compare the Compare Compare Sets of Straw Do the lengths of Compare Compare Pieces straws sum of the (b + c) and a (a + c) and b Pieces straws (m + n) and l (l + n) and m form a lengths of form a shorter straws triangle or triangle or (l + m) with that not? shorter straws not? of the longest (a + b) with that of the longest length c length c l m n YES NO l+m <,>,= n m+n <,>,= l l +n <,>,= m 1. 3 3 7 l m n YES NO l+m <,>,= n m+n <,>,= l l +n <,>,= m 2. 3 3 51. 3 > 3 10 > 3 3. 4 6 102. 3 37 √ 6 < 7 10 > 38 > 3 4. 4 6 93. 4 > 4 14 > 6 5. 5 5 104. 4 35 √ 6 >58 > 4 13 > 6 6. 5 5 85. 5 > 5 15 > 5 7. 6 7 116. 5 6 10 √ 10 = 10 16 > 5 13 > 5 8. 6 7 97. 6 > 6 17 > 7 9. 4 7 128. 6 69 √ 10 > 9 15 > 6 15 > 7 10. 4 7 109. 4 > 4 16 > 710. 4 5 10 √ 10 = 10 15 > 4 14 > 7 58 √ 10 > 8 13 7 11 √ 13 > 11 18 79 √ 13 > 9 16 7 12 √ 11 < 12 19 7 10 √ 11 > 10 17 ?E S T I O 1. Making Conjectures:1. Making Conjectures: QU NS 1.1 What pattern did you observe when you 1.1 compared the sum of the lengths of the two • If the sum of the lengths of the two shorter straws is EQUAL to the shorter straws with the length of the longest Mathematics in length of the longest side, a triangle cannot be formed. straw? Write your findings by completing the Geography • If the sum of the lengths of the two shorter straws is LESS THAN the phrases below: length of the longest side, a triangle CANNOT be formed. • If the sum of the lengths of the two shorter Feasible Possible • If the sum of the lengths of the two shorter straws is GREATER straws is equal to the length of the longest Distance THAN the length of the longest side, a triangle CAN be formed. straw ____________________________. • If the sum of the lengths of the two shorter (McDougal Little 1.2 straws is less than the length of the longest Geometry, 2001) • When the straws form a triangle, the sum of the lengths of any two straw ____________________________. straws is greater than the third straw. • If the sum of the lengths of the two shorter Suppose you know the • When the straws do not form a triangle, the sum of the lengths of any straws is greater than the length of the longest following information about two straws is less than or equal to the third straw. straw ___________________________. distances between cities in the Philippine Islands:2. Triangle Inequality Theorem 3 1.2 What pattern did you observe with the sets of • The sum of the lengths of any two sides of a triangle is greater than straws that form and do not form a triangle? Cadiz to Masbate ≈ 159 km the third side. Complete the phrases below to explain your Cadiz to Guiuan ≈ 265 km findings: • When the straws form a triangle, the sum of Considering the angle the lengths of any two straws __________. formed with Cadiz as the vertex, describe the range of possible distances from Guiuan to Masbate. 430

Answer Key to Quiz No. 3 • When the straws do not form a triangle, the sum of the lengths of any two straws__________.1. Description: • AW + EW > AE 2. Your findings in this activity describe Triangle Inequality Theorem 3. • AW + AE > EW State the theorem by describing the relationship that exists between • AE + EW > AW the lengths of any two sides and the third side of a triangle. • The sum of the lengths of any two sides of a triangle is2. ___________________. Hints In Simplified Is the Can a QUIZ No. 3 Symbols Form simplified triangle Directions: Write your answer on a separate answer sheet. form true? be 1. Describe sides AW, EW and AE of ∆AWE using Triangle Inequality Theorem 3. formed? Justify Is the sum of 8 YES YES 1 and 10 greater Is 8 + 10 > 14 Is 18 > 14 because than 14? the sum 2. Your task is to check whether it is possible to form a triangle with lengths 8, 10, Is the sum of 8 YES of any and 14. Perform the task by accomplishing the table shown. Let the hints guide 2 and 14 greater Is 8 + 14 > 10 Is 22 > 10 two sides you. than 10? is greater Is the Can a simplified triangle be Is the sum of 10 YES than the Hints In Simplified form true? 3 and 14 greater Is 10 + 14 > 8 Is 24 > 8 third side. Symbols Form formed? Justify than 8? Which question should be enough to find out if a triangle can be formed? 1 Is the sum of 8 and • The question asking whether 8 +1 0 > 14 should be enough to find out 10 greater than 14? if triangle is formed from the sides because 8 and 10 are the shorter 2 Is the sum of 8 and sides. 14 greater than 10?3. 3 Is the sum of 10 and 14 greater than 8? Is the Find out if: Simplified simplified form Can a triangle be Which question should be enough to find out if a triangle can be formed? Forms formed? Justify true? 1 5 + 8 > 13 13 > 13 NO YES because the 3. Is it possible to form a triangle with sides of lengths 5, 8, and 13? Complete the 2 5 + 13 > 8 18 > 8 table to find out the answer. 3 8 + 13 > 5 YES sum of any two 21 > 5 sides is greater than Is the simplified Can a triangle be YES the third side. Find out if: Simplified Forms form true? formed? Justify Which question should be enough to find out if a triangle can be formed? 1 • The question asking whether 8 + 10 > 14 should be enough to find out 2 if triangle is formed from the sides because 8 and 10 are the shorter sides. 3 Which question should be enough to find out if a triangle can be formed? 431

4. 4. Can you form a triangle from sticks of lengths 7, 9, and 20? Mathematics in Architecture Find out if: Simplified Is the Can a triangle be formed? 5. Find out if: Simplified Is the Can a Forms simplified Justify Forms simplified triangle be World’s Thinnest form true? form true? formed? House: Keret House by Jakub Szczesny 1 7 + 9 > 20 16 > 20 7 + 9 > 20 16 > 20 Justify 2 7 + 20 > 9 27 > 9 7 + 20 > 9 27 > 9 The four-feet-wide house is 3 9 + 20 > 7 29 > 7 9 + 20 > 7 29 > 7 1 built on a tiny space between two buildings in Warsaw, 2 Poland. 3 Questions: 1. Explain why Architect Which question should be enough to find out if a triangle can be formed? Which question should be enough to find out if a triangle • The question asking whether 7+9>20 should be enough to find out can be formed? Szczesny used triangu- lar design in building the if triangle is formed from the sides because 7 and 9 are the shorter Study the figure shown and complete the table of inequalities house? sides. using Triangle Inequality Theorem 3. 2. Why is it not advisable to use rectangular design?5. To see more photos of the CA + AR > CR house, visit this website link: http://design-milk.com/ ER + AR > AE CA + AR > worlds-thinnest-house- ER + AR > keret-house-by-jakub- AR < RE + AE < RE + AE szczesny/?fb_action_ AC + CE > ids=4022332433678&fb_ AC + CE > CE action_types=og. AE + CE likes&fb_source=aggregation&fb_ AC AE + CE aggregation_ id=2883814812375826. s1 + s2 > s37. If t is the third side then, the following should be satisfied: 7 + 10 > t 7 + t > 10 t + 10 > 7 6. Using Triangle Inequality Theorem 3, what inequality will you write to check whether segments with lengths s1, s2, and s3 form a triangle if s1 < s2 < s3? t < 7+10 t < 10 − 7 t < 7 − 10 7. If two sides of a triangle have lengths 7 feet and 10 feet, what are the possible t < 17 t<3 t < -3 integral lengths of the third side? Between what two numbers is the third side? But length should be greater than zero. The values described must be excluded. • Therefore, side t may have the following measurements in ft.: {4, 5, 8. The distance Klark walks For items no. 8-10, use the figure shown 6,…14, 15, 16} from home to school is School • Side t has lengths between 3 ft and 17 ft. 120 meters and 80 meters 8. Xylie’s estimation, 180 meters, is feasible. The distance of 180 is within when he goes to church Church the range of (120 – 80 = 40) and (120 + 80 = 200). from home. Xylie estimates9. Because S1 < 80 < 120 and 40 < S1 < 200, then 40 < S1 < 8010. Path No. 2: School to Church. Justification: Triangle Inequality Theorem 3 that the distance Klark 120 m11. Errors: walks when he goes directly 80 m • With two marks on EF, it is the longer side so the angle opposite it must also be the larger angle. However, it is opposite the shortest to Church, coming from Home angle—angle D. school, is 180 meters. Realee’s estimation is 210 meters. Which estimation is feasible? Justify your answer. 9. Supposing that the shortest distance among the three locations is the school- church distance, what are its possible distances? 432

Mathematics in Geography: Feasible Possible Distance 10. Which of the following paths to church is the shortest if you are from school? Justify your answer.From the map, it is clear that the distance d of Guiuan to Masbate is thelongest. Hence, the distance of Guiuan to Masbate must be greater than • Path No. 1: School to Home then to Church265 m but less than the sum of 159 and 265 m, which is 424 m. Therefore, • Path No. 2: School to Church265 < d < 424. 11. Some things are wrong with the measurements on the sides and angles of the triangle shown. What are they? Justify your answer.1. Architect Szczesny used a triangular design because it is enough for him The next activity is about discovering the triangle inequality to provide a bedroom, a bathroom, and a kitchen. theorem involving an exterior angle of a triangle. Before doing it, let us first recall the definition of an exterior angle of a triangle.2. Possible reasons: • Unlike a rectangular design, a triangular design has the roof already steep so rainwater or snow will just slide easily. • With a rectangular design, it needs two stands as foundations to achieve balance. In that case, where will he place the ladder to the house?Mathematics in Art: Color Triangle 1. Color combinations By extending MN of ∆LMN to a point P, MP is formed. As a • Yellow and Blue = Green result, ∠LNP forms a linear pair with ∠LNM. Because it forms a linear • Red and Yellow = Orange pair with one of the angles of ∆LMN, ∠LNP is referred to as an exterior • Blue and Red = Violet angle of ∆LMN. The angles non-adjacent to ∠LNP, ∠L and ∠M, are called remote interior angles of exterior ∠LNP. 2. Possible number of exterior angles In the triangle shown, ∠4, ∠5 and ∠6 are exterior angles. The • Two (2) remote interior angles of ∠4 are ∠2 and ∠3; of ∠5, ∠1 and ∠3; of ∠6, ∠1 • Four (4) and ∠2. • Two (2) Internet LearningNotes to the Teacher Measures of Interior and Exterior Angles ofAfter reviewing their knowledge on exterior angles of triangles, direct theirattention to one of the artworks in the activity Artistically Yours. Let them a Triangledetermine the exterior angles and their corresponding remote interior angles. Interactive:Once done, invite them to go to the internet and perform the interactive activity • h t t p : / / w w w .from http://www.mathwarehouse.com/geometry/triangles/angles/remote-exterior-and-interior-angles-of-a-triangle.php. mathwarehouse.com/ geometry/triangles/ angles/remote-exterior- and-interior-angles-of-a- triangle.php 433

Activity No. 7: Activity 7 MEASURE MANIA: Mathematics in ArtMEASURE Mania: Exterior or Remote Interior? EXTERIOR OR REMOTE INTERIOR? Color TriangleFor Activity no. 7, explain to them that Grade 8 students should have the Materials Needed: protractor, manila paper, and ruler The Color Triangle makespassion for getting measurements; hence, the title has mania in it (mania for Procedures: it easier to determine thepassion). And one has to be sure of his/her measure, hence, MEaSURE. resulting color if two colors 1. Measure the numbered angles of ∆HEY, ∆DAY, and ∆JOY. are combined.Tell the students that in this activity, they will find out the inequality that exists 2. Replicate the table in this activity on a piece of manila paper.between an exterior angle of a triangle and each of its remote interior angles. 3. Indicate the measures on your table and write your answers Questions:But before you proceed, decide on the GPE and Tolerance Interval of your 1. What is the resultingmeasurements. to the ponder questions on a piece of manila paper. color with the followingLet them get the measurement of the exterior and interior angles of the triangles, 1 Y D2 J2 6Y combinations?compare them and write their findings and answers to ponder questions on a H5 2 4 4 3 • Yellow and Bluepiece of manila paper. Process their outputs. You have to consider the GPE • Red and Yellowand the Tolerance Interval. Their answer to activity question no. 4 should be 4 15 63 12 • Blue and Redwritten on a piece of cartolina and posted on a display board of math concepts. Y A O 2. How many possible 6 exterior angles do theEnd the activity with the students answering Quiz No. 4. E3 following sets of color triangles have? • B, R, Y • G, O, V • YO, YG, RO, RV, BG, BV To read more about the color triangle, visit this website link: http://www.atpm. com/9.08/design. shtml Answer key to Activity No. 7 Questions MEASURES1. The answer to each item is: >. Name of 1st Remote Interior 2nd Remote Interior 3rd Remote Interior2. The answer to each item is: >. Triangle Exterior ∠s Exterior ∠s Exterior ∠s3. The answer to each item is: >.4. Conjecture: The measure of an exterior angle of a triangle is greater ∠ ∠4 ∠6 ∠ ∠5 ∠6 ∠ ∠4 ∠5 than the measure of either interior angle. ∠1 ∠2 ∠3 ∆HEYAnswer key of Quiz No. 4 ∆DAY1. Inequalities: Considering ∆HAM ∆JOY m∠HAT > m∠M Considering ∆REA m∠HAT > m∠H m∠CAR > m∠E m∠CAR > m∠R 434

2. QU ?E S T I ONS 1. Compare the measure of exterior ∠1 with either remote interior ∠4 or m∠AED > m∠AED ∠6 using the relation symbols >, <, or =. m∠DEB > m∠DCE • In ∆HEY, m∠1 is _____ m∠4. m∠DEB = m∠DBE • In ∆HEY, m∠1 is _____ m∠6. m∠CDE < m∠DEB • In ∆DAY, m∠1 is _____ m∠4. m∠DEB < m∠ACD • In ∆DAY, m∠1 is _____ m∠6. • In ∆JOY, m∠1 is _____ m∠4.3. • In ∆JOY, m∠1 is _____ m∠6. ∆DEB ∠AED, ∠BDF or ∠CDF ∆CDG ∠AGD, ∠CGE, ∠CDF, ∠BCG 2. Compare the measure of exterior ∠2 with either remote interior ∠5 or ∆AGE ∠CGE, ∠BEG, ∠AGD ∠6 using the relation symbols >, <, or =. ∆BAC ∠ACD or ∠GCD • In ∆HEY, m∠2 is _____ m∠5. • In ∆HEY, m∠2 is _____ m∠6. Teacher’s Note and Reminders • In ∆DAY, m∠2 is _____ m∠5. • In ∆DAY, m∠2 is _____ m∠6. Don’t • In ∆JOY, m∠2 is _____ m∠5. Forget! • In ∆JOY, m∠2 is _____ m∠6. 3. Compare the measure of exterior ∠3 with either remote interior ∠4 or ∠5 using the relation symbols >, <, or =. • In ∆HEY, m∠3 is _____ m∠4. • In ∆HEY, m∠3 is _____ m∠5. • In ∆DAY, m∠3 is _____ m∠4. • In ∆DAY, m∠3 is _____ m∠5. • In ∆JOY, m∠3 is _____ m∠4. • In ∆JOY, m∠3 is _____ m∠5. 4. Making Conjecture: Your comparison between the measure of an exterior angle of a triangle and either interior angle in this activity describes the Exterior Angle Inequality Theorem. With the pattern that you observed, state the exterior angle inequality theorem. • The measure of an exterior angle of a triangle is__________. QUIZ No. 4 Directions: Write your answer on a separate answer sheet. 1. Use the Exterior Angle Inequality theorem to write inequalities observable in the figures shown. T E A H A 83o 118o R 35o C M 51o 435

Mathematics in Psychology: Robert Sternberg’s Considering ∆REA Considering ∆HAM Triangular Theory of Love1. Possible Answer: 2. Use >, <, or = to compare the measure of angles. • At the angles of the triangles are the Liking (intimacy), Infatuation m∠AED m∠CED (passion), and Empty Love (Commitment). m∠DEB m∠DCE m∠DEB m∠DBE • The sides are made up of Romantic Love, Companionate Love, and m∠CED m∠DEB Fatuous Love. Romantic love is a result of passion and intimacy. Companionate love is a result of intimacy and commitment. Fatuous 3. Name the exterior angle/s of the triangles shown in the figure. Love is a result of passion and commitment. ∆DEB ∆CDG • Consummate Love is at the interior of the triangle and it is a result of ∆AGE passion, intimacy, and commitment. ∆BAC2. Possible Answer: All are important. However, the most important is commitment because love based on commitment will survive amidst challenges like illness and poverty.3. Possible Answer: Consummate love because it has all the elements that every human being dreams of.4. Possible Answers: You have successfully developed all the theorems on inequalities in one triangle. Activity No. 8 can be performed using them. Good luck! • Consummate love > companionate love > fatuous love > romantic love Activity 8 My Grandpa, My Model of • Commitment > intimacy > passion Healthy Lifestyle!Activity No. 8: Leruana has a triangular pictureMy Grandpa, My Model of a Healthy Lifestyle frame that her grandpa gave her on her 13th birthday. Like her, his grandpa loves triangularA day before the Activity No. 8 is scheduled to be performed, shapes. Since it is going to be his grandpa’sreproduce the desired number of copies of the Grandpa pictures 65th birthday soon, her birthday gift idea is toand the pictures of the suggested outputs of the activity. Note have two triangular frames made so she canthat answers may vary so analyze the merit of students’ outputs place in them photos of his grandpa as healthcarefully. exercise instructor. As her woodworker friend, she asks you to do the triangular frames for her. To determine the shapes of the picture frames, how should the photos be cropped? 436

Possible Answers to Activity 8: Mathematics in1. Psychology Robert Sternberg’s Triangular Theory of Love2. Any part of the body can't be cropped. Questions: QU?E S T I ONS 1. How do you plan to crop the photographs? • For the standing grandpa, the line on his back, and for the sitting one, • Indicate the vertices of the triangular part of the • his outstretched legs or the line from the tip of his foot to his head 1. Study the triangle intently. photos. What do you understand • Mark the sides of the new triangular photos.Activity No. 9: about the triangularClock Wisdom: Pretty One! theory of love? 2. What made you decide to have that shape and not something else?For Activity no. 9, it is advisable for you to bring a real clock. You have to show 2. Which mean more to • What is your basis for determining the largestto them the angles formed by the short and long hands at 1PM, 2 PM, 3PM you—passion, intimacy corner?and 4PM. or commitment? • What is your basis for determining the longest side? 3. Which love you would like to have in the future—romantic, fatuous, companionate or consummate love? 4. How do you rank romantic love, fatuous love, companionate love and consummate love using combined inequality? To help you decide, visit http://gentlemencalling. wordpress.com/2012/03/13/ for-the-love-of-triangles/You have to elicit from the class the measures of the angles formed by the Activity 9 Clock Wisdom, Pretty One!hands of the clock at the aforementioned times. Let the students determine theangle by giving them these clues: • One complete revolution is 360 degrees • The whole revolution is subdivided into 12 hoursThey must realize that each subdivision is (360/12) degrees or 30 degrees. A complete revolution around a point is equivalent to 360º. The minute and hourOnce the measures of the angles of clock faces A, B, C, and D are determined, hands of the clock also cover that in a compete revolution.let the groups do the activity and answer questions in Ponder Time. Let thempost their outputs written on manila paper. Process their outputs. Answers to Materials: ruler and manila paperquestion no. 6 and 7 (including the drawings of ∆CAT and ∆DOG) of ponder Procedure:time must be written on a piece of cartolina and posted on a display board ofmath concepts. 1. Replicate the activity table on a piece of manila paper. 2. Study the faces of the clock shown at different hours one afternoon and complete your copy of the activity table. 437

Answer Key to Questions in Activity 9 3. Write also your answers to the ponder questions on a piece of manila paper. 4. Compute for the measure of the angle formed by the hands of the clock given 1. The short hands of the clock in clock faces A, B, C, and D are equal (=). 2. The short hands of the clock in clock faces A, B, C, and D are equal (=). that one revolution for each hand is equivalent to 360°. 3. The angles formed by the hands of the clock are called as included Clock Face Time Measure of angle Distance between angles. (Exact PM Hours ) formed by the the tips of the 4. The later in the afternoon the hour is, the larger is the angle. hour hand and hour hand and 5. The measure of the distance between the tips of the hands of the clock is minute hand minute hands (in mm) influenced by the measure of the included angle at a certain time. 6. If two sides of one triangle are congruent to two sides of another triangle, A but the included angle of the first triangle is greater than the included B angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. C 7. If AC ≅ OD, AT ≅ OG, and m∠A > m∠O; then CT > DG 8. Note: Answers may vary. D 9. Some examples: Ladies’ fan, door hinge, tail of a peacock, geometric compass, puller, nipper, pliers, pages of a book, arms with the elbow joint QU?E S T I ONS 1. Write your observations on the following: as the hinge, legs with knee joint as the hinge, etc. • The lengths of the roofs at the left part of both houses __. • The Lengths of the roof at the right part of both houses __.Activity No. 10: • The lengths of the roof bases of both houses __.Roof-y Facts, Yeah! • The Roof angles of both houses __. 2. What influences the measures of the roof angles of both houses? Justify.Before starting group Activity 10, decide for the GPE and Tolerance Interval of 3. Making a Conjecture: Your findings describe the Converse of Hinge Theoremthe measurements. Proceed to the following: groups working on the activityand answering activity questions while you roam around to give technical (This is otherwise known as SSS Triangle Inequality Theorem). How will youassistance; posting of outputs; processing outputs; and writing answers of nos. state this theorem if you consider the two corresponding roof lengths as two3 and 4 (including the drawings ∆RAP and ∆YES) on a piece of cartolina to be sides of two triangles, the roof bases as their third sides, and the roof angles asposted on the display board in mathematics. included angles? State it in if-then form. If two sides of one triangle are congruent to two sides of another triangle,Let the class answer Quiz No. 5 and discuss the solutions and answers for but the third side of the first triangle is greater than the third side of the second,each item. then ________________________. 4. Using the Converse of Hinge Theorem, write an if-then statement to describe the appropriate sides and angles of ∆RAP and ∆YES. Roof Lengths Roof Lengths Lengths of HOUSE at the Right at the Left Roof Base (in Roof Angle (in cm) (in cm) cm) A B AE RY PS 5. With both houses having equal roof lengths, what conclusion can you make about their roof costs?438