Mathematics Grade 8 Part 2

Answer Key to Questions in Activity 10 Activity 10 ROOF-Y FACTS, YEAH!1. Observations: M N Q R • The lengths of the roofs at the left part of both houses are equal. L • The lengths of the roofs at the right part of both houses are equal. P House A • The lengths of the roof bases of both houses differ in lengths. Roof House B base of house A is shorter than the roof base of House B. Materials Needed: protractor, manila paper, and ruler2. The measures of roof angles are affected by the length of the roof bases. Procedure: Study the house models and complete your copy of the activity table. For If the roof base is longer, the roof angle is also larger. ponder questions, write your answers on a piece of manila paper.3. If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the HOUSE second, then the included angle of the first triangle is larger than the included angle of the second triangle. A4. If AR ≅ EY, AP ≅ ES, and PR > SY; then m∠A > m∠E. B5. Roof costs for House A is the same as roof costs for House B. QU ?E S T I ONS 1. Write your observations on the following: Teacher’s Note and Reminders • The lengths of the roofs at the left part of both houses __. • The Lengths of the roof at the right part of both houses __. Don’t • The lengths of the roof bases of both houses __. Forget! • The Roof angles of both houses __. 2. What influences the measures of the roof angles of both houses? Justify. 3. Making a Conjecture: Your findings describe the Converse of Hinge Theorem (This is otherwise known as SSS Triangle Inequality Theorem). How will you state this theorem if you consider the two corresponding roof lengths as two sides of two triangles, the roof bases as their third sides, and the roof angles as included angles? State it in if-then form. If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is greater than the third side of the second, then ________________________. 4. Using the Converse of Hinge Theorem, write an if-then statement to describe the appropriate sides and angles of ∆RAP and ∆YES. AE RY PS 5. With both houses having equal roof lengths, what conclusion can you make about their roof costs? 439

Answer Key to Quiz No. 5 QUIZ No. 5A. Directions: Write your answer on a separate answer sheet. Statements Justification A. Use the symbol <, > or = to complete the statements about the figure shown. Justify 1. If AC ≅ AD and m∠1 = m∠2, then BC = BD Hinge Theorem your answer. C 2. If BC ≅ BD and AC > AD ,then m∠4 > m∠3 Converse of Hinge 1 4 Theorem 2 3 A B 3. If AD ≅ AC and m∠2 < m∠1,then BD < BC Hinge Theorem 4. If BD ≅ BC and AD> AC ,then m∠3 > m∠4 Converse of Hinge D Theorem Statements Justification 1. If AC ≅ AD and m∠1 = m∠2, then BCB. 2. If BC ≅ BD and AC > AD ,then m∠4 BD 3. If AD ≅ AC and m∠2 < m∠1,then BD m∠3 GIVEN FACTS FOR CONCLUSION JUSTIFICATION 4. If BD ≅ BC and AD> AC ,then m∠3 MARKINGS BC m∠41. R N BY = AT BR = AN m∠B > m∠A RY > NT Hinge Theorem BA YT B. Make necessary markings on the illustration based on the given. What conclusion2. R N ∠N is not an can you make, if there is any, given the facts about the two triangles? Provide included angle BR = AT RY = NT m∠R > m∠N None justifications to your conclusions. R N BA YT3. RY > NT RN m∠B > m∠A Converse of BA BY = AT BR = AN Hinge Theorem YT BA YT FOR MARKINGS4 RN GIVEN FACTS CONCLUSION JUSTIFICATION BR = AN RY = NT BY > AT BA m∠R > m∠N Hinge Theorem 1. RN YT BY = AT BR = AN m∠B > m∠A5. R N Converse of BA Hinge Theorem YT RY = NT BY = AN m∠N < m∠Y A AT > BR 2. RN B BR = A T RY = NT m∠R > m∠N YT BA YTC. 1. AC > DF 3. m∠RAT > m∠YAT 4. m∠FAE > m∠MAE 2. HI > GI 3. R ND. 2m − 1 > m + 4. Therefore, m > 5. BY = AT BR = AN RY > NT BA YTE. Sample Answer: 4 RN The hinge of the compass makes it possible to adjust the distance BR = AN RY = NT BY > AT BA between the tips of the compass point and the pencil point. Adjustments YT determine the desired lengths of radii for circles to be drawn. 5. RN RY = NT BY = AN m∠N < m∠Y BA YT 440

E. Mathematics in Fashion: Ladies’ Fan C. Using Hinge Theorem and its converse, write a conclusion about each figure. 1. It is important when it is hot and there is no air conditioning unit like in 1. F A 3. R churches. 2. When the fan is not opened completely, the distance between the tips B A T of the side frame of the fan is shorter than when the fan is opened Y E completely. D EC From the prior investigations, we have discovered the following theorems on triangle inequalities: 2. I 4. A Inequalities in One Triangle: G H M J F Triangle Inequality Theorem 1 (Ss → Aa) If one side of a triangle is longer than a second side, then the angle D. Using Hinge Theorem and its converse, solve for the possible values of m. opposite the first side is larger than the angle opposite the second side. m+4 2m − 1 Triangle Inequality Theorem 2 (Aa → Ss) If one angle of a triangle is larger than a second angle, then the side 53 3 5 opposite the first angle is longer than the side opposite the second angle. E. Enrichment Activities Triangle Inequality Theorem 3 (S1 + S2 > S3) The sum of the lengths of any two sides of a triangle is greater than the 1. Hinges in Tools and Devices length of the third side. Hinges are used to fasten two things together and allow adjustment, rotation, twisting or pivoting. Choose at least one of the following hinged devices and Exterior Angle Inequality Theorem explain how it works. The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle Inequalities in Two Triangles: Hinge Theorem or SAS Inequality Theorem If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second. Converse of Hinge Theorem or SSS Inequality Theorem: If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second.441

Explain to the students that the next activities of the PROCESS section are on 2. Mathematics in Fashion: Ladies’ Fanwriting proofs of the theorems on inequalities in triangles. From the sixteenth century up to the late 1800s throughout the whole of Europe, eachInspire the students to think clearly and systematically together as a group. fashionable lady had a fan and becauseExplain to them that in writing the proofs of theorems, focus and collaboration of its prominence, it was considered as aare the instruments for their success. “woman’s scepter”—tool for communicating her thoughts. How can we prove these theorems? http://www.victoriana.com/Fans/historyofthefan.html Writing proofs is an important skill that you will learn in geometry. It will develop your observation skills, deductive thinking, logical reasoning, and Questions: mathematical communication. Guide questions are provided to help you 1. Do you think that fan is an important fashion item? succeed in the next activities. 2. Describe the concept of inequality in triangles that is evident about a ladies’ fan. From the prior investigations, we have discovered the following theorems on triangle inequalities: In writing proofs, you have to determine the appropriate statements and Inequalities in One Triangle: give reasons behind these statements. There are cases when you only have to complete a statement or a reason. Make use of hints to aid you in your Triangle Inequality Theorem 1 (Ss → Aa) thinking. If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side. Be reminded that theorems may be proven in different ways. The proofs that follow are some examples of how these theorems are to be proven. Triangle Inequality Theorem 2 (Aa → Ss) If one angle of a triangle is larger than a second angle, then the side opposite the first For activity 11-16, you are required to use a piece of manila paper for angle is longer than the side opposite the second angle. each proof. Triangle Inequality Theorem 3 (S1 + S2 > S3)Make sure that a day before the activities in writing proofs are scheduled, The sum of the lengths of any two sides of a triangle is greater than the length ofgroups already have enough number of pieces of manila paper for the activity the third side.where tables for statements and reasons are already prepared. Exterior Angle Inequality TheoremYou may opt to let the students prepare metastrips (each piece is 1/3 of bond The measure of an exterior angle of a triangle is greater than the measure of eitherpaper cut lengthwise) and pentel pen or ball pen so that they only have to write remote interior angleeach statement or reason on a metastrip and attach it on the appropriate rowand column. Inequalities in Two Triangles:Your technical assistance is crucial in the proof-writing activities so roam Hinge Theorem or SAS Inequality Theoremaround purposely. Most of your assistance involves your directing them to If two sides of one triangle are congruent to two sides of another triangle, but therefer the review points in this module. included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second. Converse of Hinge Theorem or SSS Inequality Theorem: If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second.442

Answer Key to Activity 11: How can we prove these theorems?Proving Triangle Inequality Theorem 1 Writing proofs is an important skill that you will learn in geometry. It will develop your observation skills, deductive thinking, logical reasoning, and mathematical communication.A. Reasons Guide questions are provided to help you succeed in the next activities. Statements1. LM ≅ LP By construction In writing proofs, you have to determine the appropriate statements and give reasons2. ∆LMP is isosceles Definition of Isosceles Triangle behind these statements. There are cases when you only have to complete a statement or Base angles of isosceles triangles a reason. Make use of hints to aid you in your thinking.3. ∠1 ≅ ∠2 are congruent. Angle Addition Postulate Be reminded that theorems may be proven in different ways. The proofs that follow are4. ∠LMN ≅ ∠1 + ∠3 some examples of how these theorems are to be proven.5. ∠LMN > ∠1 Property of Inequality For activity 11-16, you are required to use a piece of manila paper for each proof.6. ∠LMN > ∠2 Substitution Property Activity 11 PROVING TRIANGLE INEQUALITY THEOREM 17. ∠2 + ∠MPN = 180 Linear Pair Postulate Triangle Inequality Theorem 1 (Ss → Aa)8. ∠MPN + ∠N + ∠3 = 180 The sum of the interior angles of a If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the triangle is 180. angle opposite the second side.9. ∠2 + ∠MPN = ∠MPN + ∠N + ∠3 Substitution/Transitive Property10. ∠2 = ∠N + ∠3 Subtraction Property11. ∠2 > ∠N Property of Inequality12. ∠LMN > ∠N Transitive Property Given: ∆LMN; LN > LM Prove: ∠LMN > ∠LNMTeacher’s Note and Reminders Proof: There is a need to make additional constructions to prove that ∠LMN > ∠LNM. With compass point on L and with radius LM, mark a point P on LN and connect M and P with a segment to form triangle. Don’t Statements Reasons Forget! 1. How do you describe the By construction relationship between LM and LP? Definition of Isosceles Triangle 2. Based on statement 1, what kind of a triangle is ∆LMP? 3. Based on statement 1, how do you describe ∠1 and ∠2? Converse of Isosceles Triangle Theorem 443

Teacher’s Note and Reminders 4. Study the illustration and write Angle Addition Postulate a statement about ∠LMN if the Don’t reason is the one given. Forget! 5. Basing on statement 4, write an Property of Inequality inequality statement focusing on ∠1. 6. Using statement 3 in statement 5: Substitution Property ∠LMN > ∠2 7. Study the illustration and write Sum of the interior angles of a triangle an operation statement involving is 180o. ∠MPN, ∠N, and ∠3 8. Study the illustration and write an Linear Pair Theorem operation statement involving ∠2 and ∠MPN 9. ∠2 + ∠MPN ≅ ∠MPN + ∠N + ∠3 What property supports the step wherein we replace the right side of statement 8 with its equivalent in statement 7? 10. What will be the result if ∠MPN is deducted away from both sides of statement 9? 11. Basing on statement 10, write an Property of Inequality inequality statement focusing on ∠N. 12 Based on statement 6 and 11: If Property of Inequality ∠LMN > ∠2 and ∠2 > ∠N, then Congratulations! You have contributed much in proving Triangle Inequality Theorem 1. In the next activity, you will see that Triangle Inequality Theorem 1 is used in proving Triangle Inequality Theorem 2. 444

Answer Key to Activity 12: Activity 12 INDIRECT PROOF OF TRIANGLE Indirect Proof of Triangle Inequality Theorem 2 INEQUALITY THEOREM 2A. Triangle Inequality Theorem 2 (Aa→Ss) If one angle of a triangle is larger than a second angle, then the side opposite the first Statements Reasons angle is longer than the side opposite the second angle. 1. MN ≯ LM such that 1. Temporary Assumption MN = LM or MN < LM Given: ∆LMN; m∠L > m∠N L 2. Considering MN = LM: Prove: MN > LM If MN = LM, then ∆LMN is an 2. Definition of isosceles triangles isosceles triangle. Indirect Proof: Consequently, ∠L = ∠N. Base angles of isosceles triangles Assume: MN ≯ LM M N are congruent. Statements Reasons The assumption that MN = LM is The conclusion that ∠L ≅ ∠N Assumption that MN ≯ LM contradicts the given that m∠L > m∠N. 1. MN = LM or MN < LM 1. false. Definition of 2. Considering MN ≅ LM : If MN ≅ 3. Considering MN < LM: 3. Triangle Inequality Theorem 1 If MN < LM, then m∠L < m∠N. (Ss  Aa) LM, then 2. The Assumption that MN < LM is The conclusion that m∠L < m∠N Consequently, what can you say contradicts the given that m∠L > m∠N about ∠L and ∠N? False of isosceles The Assumption that MN ≅ LM is triangles are congruent. 4. Therefore, MN > LM must be 4. The assumption that MN ≯ LM True False True contradicts the known fact that The conclusion that ∠L ≅ ∠N m∠L > m∠N. the given that m∠L > m∠N. Teacher’s Note and Reminders 3. Considering MN < LM: 3. Base angles of isosceles triangles If MN < LM, then are congruent. The Assumption that MN < LM is The conclusion that m∠L < m∠N True False contradicts the given that 4. Therefore, MN > LM must be 4. The that True False MN ≯ LM contradicts the known fact that m∠L > m∠N. Don’t Forget! Amazing! You have helped in proving Triangle Inequality Theorem 2. Let us proceed to prove Triangle Inequality Theorem 3 using a combination of paragraph and two-column form. You will notice that Triangle Inequality Theorem 2 is used as reason in proving the next theorem. 445

Answer Key to Activity 13: Activity 13 PROVING TRIANGLEProving Triangle Inequality Theorem 3 INEQUALITY THEOREM 3 Statements Reasons Triangle Inequality Theorem t3w(oSs1 i+deSs2 > Sa3)triangle is greater than the length of the third1. LP = LN By construction The sum of the lengths of any of2. ∆LNP is an isosceles triangle. Definition of isosceles triangle side. Base angles of isosceles triangle3. ∠LNP ≅ ∠LPN are congruent. Given: ∆LMN where LM < LN < MN Reflexive Property4. ∠LPN ≅ ∠MPN Transitive Property Prove: MN + LN > LM5. ∠LNP ≅ ∠MPN Angle Addition Postulate MN + LM > LN6. ∠MNP ≅ ∠LNM + ∠LNP Substitution Property7. ∠MNP ≅ ∠LNM + ∠MPN Property of Inequality LM + LN > MN8. ∠MNP > ∠MPN Triangle Inequality Theorem 2 (AaSa) Proof: Notice that since MN > LN and that MN > LM, then9. MP > MN Segment Addition Postulate • it’s obvious that MN + LM > LN and MN + LN > LM Substitution Property are true.10. LM + LP = MP Substitution Property • Hence, what remains to be proved is the third11. LM + LP > MN statement: LM + LN > MN12. LM + LN > MNTeacher’s Note and Reminders Let us construct LP as an extension of LM such that L is between M and P, LP ≅ LN and ∆LNP is formed. Statements Reasons By construction 1. Write a statement to describe LP and LN. 1. Don’t 2. Describe ∆LNP. 2. Bases of isosceles triangles are Forget! 3. Describe ∠LNP and ∠LPN 3. congruent. Reflexive Property of Equality 4. The illustration shows that 4. ∠LPN ≅ ∠MPN Transitive Property of Equality 5. If ∠LNP ≅ ∠LPN (statement 3) and ∠LPN ≅ ∠MPN (statement 4), then 5. 6. From the illustration, 6. ∠MNP ≅ ∠LNM + ∠LNP 446

Answer Key to Activity 14: 7. Using statement 5 in statement 6, 7.Proving the Exterior Angle Inequality Theorem ∠MNP ≅ ∠LNM + ∠MPN 8. From statement 7, ∠MNP > ∠MPN 8. Property of Inequality Triangle Inequality Theorem 2 Statements Reasons 9. Using statement 8 and the1. LQ ≅ NQ; MQ ≅ QR 1. By construction2. ∠3 ≅ ∠4 2. Vertical Angles are congruent. illustration, write a statement with 9. 3. ∆LQM ≅ ∆NQR 3. SAS Triangle Congruence the reason given.4. ∠MLN ≅ ∠1 Postulate 4. Corresponding parts of 10. From the illustration, what operation5. ∠LNP ≅ ∠1 + ∠2 involving LM and LP can you write? 10. Segment Addition Postulate6. ∠LNP > ∠1 congruent triangles are7. ∠LNP > ∠MLN congruent 11. Write a statement using statement 5. Angle Addition Postulate 10 in statement 9 11. Substitution Property of Inequality 6. Property of Inequality 12. Write a statement using statement 1 7. Substitution Property of Equality in statement 11 12. Substitution Property of EqualityTeacher’s Note and Reminders Hurray! Triangle Inequality Theorem 3 is already proven. Let us proceed to writing the proof of Exterior Angle Inequality Theorem. Activity 14 PROVING THE EXTERIOR ANGLE INEQUALITY THEOREM Exterior Angle Inequality Theorem The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle Given: ∆LMN with exterior angle ∠LNP L P Prove: ∠LNP > ∠MLN M N Don’t Proof: L R Forget! Let us prove that ∠LNP > ∠MLN by construct- Q ing the following: 1. midpoint Q on LN such that LQ ≅ NQ N P 2. MR through Q such that MQ ≅ QR M 447

Answer Key to Activity 15: Statements ReasonsProving the Hinge Theorem 1. LQ ≅ NQ; MQ ≅ QR 1. 2. What relationship exists between ∠3 and ∠4? Statements Reasons 2.1. CN ≅ CH + HN2. CN ≅ CH + WH 1. Segment Addition Postulate 3. Basing on statements 1 and 2,3. In ∆CHW, CH + WH > CW 2. Substitution Property of Equality describe two triangles from the 3. What triangle congruence postulate illustration:4. CN > CW 3. Triangle Inequality Theorem 3: supports statement 3? The sum of any two sides of a5. CN > LT triangle is greater than the third 4. Basing on statement 3, 4. side. ∠MLN ≅ 4. Substitution Property of Equality 5. Basing on the illustration, 5. Angle Addition Postulate (Using statement 2 in 3) ∠LNP ≅ 5. Substitution Property of Equality 6. Basing on statement 5, ∠LNP > ∠1 6. (Using statement in construction 1 in statement 4) 7. Using statement 4 in statement 6, 7. Substitution PropertyTeacher’s Note and Reminders Indeed, the measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. Activity 15 PROVING THE HINGE THEOREM Hinge Theorem or SAS Triangle Inequality Theorem If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second. Don’t Given: ∆CAN and ∆LYT; CA ≅ LY, AN ≅ YT, ∠A > ∠Y L T Forget! Prove: CN > LT C A N Y 448

Answer Key to Activity 16: Proof: CWIndirect Proof of the Converse of Hinge Theorem 1. Construct AW such that : • AW ≅ AN ≅ YT • AW is between AC and AN, and • ∠CAW ≅ ∠LYT. Statements Reasons H1. ∠D ≅ ∠U or ∠D < ∠U 1. Assumption that ∠D ≯ ∠U AN2. Considering ∠D ≅ ∠U: 2. SAS Triangle Congruence Consequently, ∆CAW ≅ ∆LYT by SAS Triangle Congruence Postulate. So, CW ≅ It’s given that OD ≅ LU, DG ≅ Postulate LT because corresponding parts of congruent triangles are congruent. UV. 2. Construct the bisector AH of ∠NAW such that: If ∠D ≅ ∠U, then ∆ODG ≅ ∆LUV. • H is on CN • ∠NAH ≅ ∠WAHFrom the congruence, Corresponding parts of congruent OG ≅ LV triangles are congruent Consequently, ∆NAH ≅ ∆WAH by SAS Triangle Congruence Postulate because AH ≅ AH by reflexive property of equality and AW ≅ AN from construction no. 1. So,The Assumption that ∠D ≅ ∠U is OG ≅ LV contradicts the given that WH ≅ HN because corresponding parts of congruent triangles are congruent. false. OG > LV3. Considering ∠D < ∠U: 3. SAS Inequality Theorem or If ∠D < ∠U, then OG < LV. Hinge TheoremThe assumption that ∠D < ∠U is OG < LV contradicts the given that Statements Reasons false. OG > LV 1. From the illustration: 1. CN ≅ CH + HN4. Therefore, ∠D > ∠U must be 4. Assumption that ∠D ≯ ∠U is 2. CN ≅ CH + WH 2.true. proven to be false. 3. In ∆CHW, CH + WH > CW 3. After proving the theorems on inequalities in triangles, you are now highly 4. Using statement 2 in 3: 4.equipped with skills in writing both direct and indirect proofs. Moreover, you CN > CWnow have a good grasp on how to write proofs in paragraph and/or two-columnform. 5. Using statement in construction 1 in 5. You will be undergoing more complex application problems involving statement 4: CN > LTinequalities in triangles in the next section. Dear Concept Contractor, your task is to revisit your concept museum. Bravo! The Hinge Theorem is already proven. Notice that the use of paragraph formHow many more tasks can you tackle? Which concepts you have built on the first part of the proof of the Hinge Theorem shortens the proof process.previously need revision? Check also your decisions in Activity No.1. Wouldyou like to change any decision? Activity 16 INDIRECT PROOF OF THE CONVERSE How can you justify inequalities in triangles? Do you have a newinsight on how to address this essential question raised in the activity OF HINGE THEOREMArtistically Yours? Now that you know the important ideas about this topic, let’s go deeper Converse of Hinge Theorem or SSS Triangle Inequality Theoremby moving on to the next section. If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second. 449

Appreciate the students’ interest and diligence in tackling the proof-writing Given: ∆ODG and ∆LUV; UDactivities. Praise is a great form of motivation. OD ≅ LU, DG ≅ UV, OG > LVAt this point, let them revisit their answers in Activities No. 1, 2, and 3. Prove: ∠D > ∠UWHAT TO REFLECT AND UNDERSTAND: LThe REFLECT AND UNDERSTAND section exhibits activities designed to Ointensify the students’ understanding of the lesson. These activities are mostlyon solving non-routine problems, writing proofs, and wrapping up of all the Indirect Proof: V Gconcepts and skills learned in the lesson. Assume: ∠D ≯ ∠U ReasonsLike in the Process/Do section, your task in this section is to manage group Statementswork in the problem-solving activities. It is suggested that you have to play 1. Assumption thatan active role in the discussion of the solutions of the model problems so that 1. ∠D ≅ ∠U and ∠D < ∠Ustudents will have an extensive grasp on the thinking processes undertaken insolving the problems. 2. Considering ∠D ≅ ∠U: 2. Triangle Congruence It’s given that OD ≅ LU, DG ≅ UV.Your guidance and supervision of the students in the It’s-Your-Turn problem If ∠D ≅ ∠U, then ∆ODG ≅ ∆LUV. Postulatesolving activities of this section lead to the attainment of the following goals: OG ≅ LV • solve problems that require application of the knowledge and skills in inequalities in triangles; The Assumption that ∠D ≅ ∠U is false. • solve problems that require writing proofs; and • unlock all the triangles in their concept museum. 3. Considering ∠D < ∠U: If ∠D < ∠U, 3. Hinge Theorem thenDiscuss comprehensively the solutions and answers to It’s-Your-Turn problemsenhance their understanding and reinforce their learning. OG < LV contradicts the given that OG > LV 4. 4. Assumption that ∠D ≯ ∠U is proven to be false. After proving the theorems on inequalities in triangles, you are now highly equipped with skills in writing both direct and indirect proofs. Moreover, you now have a good grasp on how to write proofs in paragraph and/or two-column form.For Activity No. 19, invite them to make a research on the task under Career in You will be undergoing more complex application problems involving inequalities inMathematics—Air Traffic Controller. triangles in the next section. Dear Concept Contractor, your task is to revisit your concept museum. How many more tasks can you tackle? Which concepts you have built previously need revision? Check also your decisions in Activity No.1. Would you like to change any decision? How can you justify inequalities in triangles? Do you have a new insight on how to address this essential question raised in the activity Artistically Yours? Now that you know the important ideas about this topic, let’s go deeper by moving on to the next section.450

Answer Key to Activity 17: WWhhaatt ttoo UUnnddeerrssttaanndd Show Me the Angles!!!Answer key to Activity No. 17 Watch-This Questions! Having developed, verified, and proved all the theorems on triangle inequalities1. The value of x is solved first because knowing its value leads to in the previous section, your goal now in this section is to take a closer look at some aspects of the topic. This entails you to tackle on more applications of the theorems on determining the values of the angles of the triangular frame. triangle inequalities.2. The sum of the angles is equated to 180 because the sum of the angles Your goal in this section is to use the theorems in identifying unknown inequalities of a triangle is always 180 degrees. in triangles and in justifying them.3. Triangle Inequality Theorem 2 (AaSs)4. Even without the actual measurements, we are sure that our answer is The first set of activities showcases model examples that will equip you with ideas and hints on how to conquer problems of the same kind but already have twists. correct because we have used the theorems we developed, verified and When it is your turn to answer, you have to provide justifications to every step you proved. take as you solve the problem. The model examples provide questions for you to answer. Your answers are the justifications.Note: Explain to the students that when we justify our answer using theoremsor postulates, we are justifying deductively. So, deductively, it is sure that the The second set of activities requires you to use the theorems on inequalities in triangles in solving problems that require you to write proofs.answer is correct. There are no limits to what the human imagination can fathom and marvel. FunIt’s Your Turn! and thrill characterize this section. It is also where you will wrap up all the concepts you learned on Triangle Inequalities.m∠S + m∠E + m∠A = 180 m∠E = 2x − 1 m∠A = 4x − 3 = 4(21) − 3 Activity 17 SHOW ME THE ANGLES!!!58 + (2x − 1) + (4x − 3) = 180 = 2(21) − 1 = 84 − 3 = 8158 + 2x + 4x − 1 − 3 = 180 = 42 − 1 6x + 54 = 180 = 41 6x = 180 − 54 6x = 126 Watch this! For extra fun, groups of students in a class are tasked to create algebraic expressions x = 21 to satisfy the measures of the angles of their triangular picture frame project. If the measureSince ∠A > m∠E > m∠S, then the longest side is opposite ∠A, ES, and the of the angles are as follows: m∠A = 5x – 3, m∠C = 2x + 5, m∠E = 3x – 2, arrange the sides ofshortest side is opposite ∠S, AE. the frame in increasing order. Solution: To solve for x: Solving for Solving for Solving for m∠A m∠C m∠E (5x - 3) + (2x + 5) + (3x – 2) = 180 m∠A= 5x – 3 m∠C = 2x + 5 m∠E = 3x – 2 5x + 2x + 3x – 3 + 5 – 2 = 180 = 5(18)–3 = 2(18) + 5 = 3(18)–2 10x – 5 + 5 = 180 = 90 – 3 = 36 + 5 = 54 – 2 10x = 180 = 87 = 41 = 52 x = 18 Therefore, listing the sides in increasing order should follow this order: Sides opposite ∠C, ∠E, and ∠A. That is, AE, AC, and CE. 451

Answer Key to Activity 18: QU ?E S T I ONS 1. Why is the value x being solved first? Believe Me, There are Lots of Possibilities! 2. Why is the sum of the angles being equated to 180°? 3. What theorem justifies the conclusion that the increasing order of1. Triangle Inequality Theorem 3(S1 + S2 > S3)2. Even without actually drawing all the possible lengths of the third side to the sides is AE, AC, and CE? 4. What makes us sure that our answer is correct considering that we form a triangle with known sides 11 and 17, we are convinced that our answer is correct because its basis, Triangle Inequality Theorem 3, is a have not exactly seen the actual triangle and have not used tools to theorem that we have developed, verified and proved. Deductively, we measure the lengths of its sides and the measures of its angles? are convinced that our answer is correct.3. Relationship: 6 is the difference when 17 is subtracted from 11. It’s Your Turn!4. Relationship: 28 is the sum of 11 and 17.5. l – s < t < l + s Angle S of the triangular picture frame of another group is 58°.6. There is an infinite number of possible lengths for the third side t. The rest of the angles have the following measures: m∠E = 2x – 1,Note: Remind them of their lesson on the set of rational numbers or fractions m∠A = 4x – 3. Determine the longest and the shortest side. Givebetween 0 and 1 in Grade 7. Because a fraction between 0 and 1 can be in justifications.the form 1/M, M can be any value greater than 1. Hence, M can be 1, 000,000 or more. Thus, there are infinite fractions between 0 and 1. Notice that Activity 18 BELIEVE ME, THERE ARE LOTS OF POSSIBILITIES!this concept is also applicable to lengths between 6 and 28.It’s Your Turn! Watch this! Problem:The lengths of the sides of a triangle are 16 – k, 16 and 16 + k. What is thepossible range of values of k? You are tasked to draw a triangle wherein the lengths of two sides are specified. What are the possible lengths for the third side of the triangle you will draw if two sides should be 11 and 17, respectively? How many possible integer lengths has the third side?Using the Triangle Inequality Theorem 3, let us find the range of values for k: Solution: Since the third side is unknown, let’s represent its length by t.16 + (16+ k) > 16 (16 – k) + 16 > 16 + k Inequality 1 Inequality 2 Inequality 316 + 16 + k > 16 16 – k + 16 > 16 + k 32 – k > 16 + k 11 + 17 > t 11 + t >17 17 + t >11 32 + k > 16 32 – 16 > k + k 28 > t t > 17 – 11 t > 11 – 17 k > 16 – 32 16 > 2k t < 28 t>6 t>–6 k > -16 8 > k k < 8 t must be less than 28 t must be greater than 6 Values of t to be -16 < k disregarded The resulting inequalities show that t must be between 6 and 28, that is, 6 as the Writing them as a combined inequality, the answer is -16 < k < 8. lower boundary and 28 as the higher boundary. Using combined inequality, the order by which they will be written should be 6, t, then 28. Therefore, • the possible lengths for the third side is 6 < t < 28. • the set of possible integer lengths for the third side of the triangle is described as follows: {7, 8, 9, …,27}. Hence, there are 27 – 6 = 21 possible integer lengths for the third side. 452

Answer Key to Activity 19: QU ?E S T I ONS 1. What theorem justifies the three inequalities being written about the And You Thought Only Surveyors Trace, Huh! sides?1. 110 and 70 degrees form a linear pair, also 90 and 90 degrees. 2. Are you convinced that 6 < t < 28 is accurate even if you have not tried2. Triangle Inequality Theorem 2 (AaSs) drawing all the possible lengths of the third side to form a triangle with3. Probably yes. However, solving it is easier through a detailed illustration. 11 and 17? Why? Solving it mentally would be tedious because there are plenty of 3. Do you observe a relationship existing between 6 in 6 < t < 28 and the information in the problem two known lengths 11 and 17? Describe the relationship.4. Had we not seen the illustration and had we not known about Triangle Inequality Theorem 2, the answer would be that their distance from the 4. Do you observe a relationship existing between 28 in 6 < t < 28 and centre of the oval is the same because they have travelled the same the two known lengths 11 and 17? Describe the relationship. length from start to where they stop.5. We are convinced that the conclusion is true because the basis is Triangle 5. If the known lengths are l and s, where l is longer and s is shorter, what Inequality Theorem 2 that we developed, verified and proved to be true. should be the formula in solving for the unknown third side t? Deductively, we are convinced that the conclusion is true. 6. There are 21 possible integer lengths for the third side when twoIt’s Your Turn! respective sides of a triangle have lengths 11 and 17. Can you count all the possible lengths other than the integer lengths? Explain.The lengths of the sides of a triangle are 16 – k, 16 and 16 + k. What is thepossible range of values of k? It’s Your Turn! Problem:Using the Triangle Inequality Theorem 3, let us find the range of values for k: The lengths of the sides of a triangle are 16 – k, 16, and 16 + k. What is the range of the possible values of k? Create a table of the possible integer lengths of the sides of the triangle. Is 16-k always the shortest length? Develop a general formula for lengths with this description. Provide justifications. Activity 19 AND YOU THOUGHT ONLY SURVEYORS TRACE, HUH!1. 500 Shielou Watch this! Problem: 1300 Kerl and Kyle play with their roller skates at the town oval. From the centre of the oval, Kerl skates 4 meters east and then 5 meters south. Kyle skates 5 meters west. He then takes a right turn of 70° and skates 4 meters. Who is farther from the centre of the oval? Therefore, Chloe is farther from the rotunda. Solution: Kyle Justification is the Hinge Theorem. 4 km 4 km 70o 110o 90o 1450 5 km 350 5 km Chloe Therefore, Kyle is farther than Kerl from the center of the oval. Kerl 453

2. Enrichment Activity ?E S T I O 1. How are 110° and 90° produced? 2. What theorem justifies the conclusion that Kyle is farther than KerlCareer in Mathematics: Air Traffic Controller QU NS from the center of the oval?Sample Research: 3. Would this problem be answered without a detailed illustration of thehttp://www.nexuslearning.net/books/ml-geometry/Chapter5/ML%20Geometry%205-6%20Indirect%20Proof%20and%20Inequalities%20 problem situation? Explain.in%20Two%20Triangles.pdf 4. Had the illustration of the problem not drawn, what would have beenYou and a friend are flying separate planes. You leave the airport and fly your initial answer to what is asked? Explain.120 miles due west. You then change direction and fly W 30° N for 70 5. We have not actually known Kerl and Kyle’s distances from the centermiles. (W 30° N indicates a north-west direction that is 30° north of duewest.) Your friend leaves the airport and flies 120 miles due east. She of the oval but it is concluded that Kyle is farther than Kerl. Are youthen changes direction and flies E 40° S for 70 miles. Each of you has convinced that the conclusion is true? Explain.flown 190 miles, but which plane is farther from the airport? It’s Your Turn!SOLUTION 1. Problem: From a boulevard rotunda, bikers Shielou and Chloe who haveBegin by drawing a diagram, as shown below. Your flight is representedby ∆PQR and your friend’s flight is represented by ∆PST. uniform biking speed, bike 85 meters each in opposite directions— Shielou, to the north and Chloe, to the south. Shielou took a rightYou N turn at an angle of 50o and Chloe, a left turn at 35o. Both continue R WE biking and cover another 60 meters each before taking a rest. Which biker is farther from the rotunda? Provide justifications.70 mi 150o airport 120 mi S S Q 120 mi P 2. Enrichment Activity Career in Mathematics: Air Traffic Controller 140o 70 mi Air traffic controllers coordinate the movement T your friend of air traffic to make certain that planes stay a safe distance apart. Their immediate concernBecause these two triangles have two sides that are congruent, you can is safety, but controllers also must directapply the Hinge Theorem to conclude that RP is longer than TP. planes efficiently to minimize delays. Therefore, your plane is farther from the airport than your friend’s plane. They must be able to do mental math quickly and accurately. Part of their job is directing aircraft at what altitude and speed to fly. Task: Make a research of problems related to the work of air traffic controllers. Solve it and present it in class 454

Answer Key to Activity 20: Activity 20 TRUST YOURSELF, YOU’RE A GEOMETRICIAN! Trust Yourself, You’re a Geometrician!1. Side HT is the longest because triangle HIT looks bigger in the figure. Watch this! F The diagram is not drawn to scale. Note: Explain to the students that the problem says that the diagram is Problem: A I Which of the lengths HF, HA, HI, not drawn to scale so answers based on the drawing without considering T 7o and HT of polygon FAITH is the the given data would be faulty. It is advised that you integrate it with this 8o saying: Do not judge a book by its cover—that outward appearances can Solution: F 9o longest? Which is the shortest? be misleading.2. It is necessary to consider each right triangle individually because a A 83o 7o H side of one triangle is also a side to another triangle—the triangles have 8o common sides. 82o 9o Considering ∆HIT:3. Triangle Inequality Theorem 2 (AaSs) I HT < HI4. Deductively, we are convinced that the conclusion is true. 81o Considering ∆HAI: T HI < HA H Considering ∆HFA: HA < HFAnswer Key to It’s Your Turn! Problem HT < HI < HA < HF1. Side HT is the longest because triangle HIT looks bigger in the figure can Therefore, the longest side is HF and the shortest side is HT. be misleading.2. It is necessary toAT=13, MT=14, MA=15 ME=9, EA=12, AM=15 Considering Exterior QU ?E S T I ONS 1. By just looking at the original figure, which side do you think is theFor angles opposite them For angles opposite them 1 > T or T < 1 longest? There is a misconception to explain why HT would have been the initial choice as having the longest side. Explain. M < MAT < T 2< M< 1 2. Why is it necessary to consider each right triangle in the figure 2 < M < MAT < T < 1 individually? Therefore, the answer is as follows: 3. What theorem justifies the choice of the longer side in each triangle? 4. Notice that the diagram is not drawn to scale. However, we are still able to tell which side is the longest and which side is the shortest. Are you convinced that your answer is true? Explain. ∠2 < ∠M < ∠MAT < ∠T < ∠1 It’s Your Turn! Problem: M9 E5 T The diagram is not drawn to scale. Using 15 1 ∠1, ∠2, ∠T, ∠M, and ∠MAT, complete the combined inequalities below: 12 <<< 13 2 A 455

Answer Key to Activity 21: Activity 21 I BELIEVE I CAN FLY I Believe I can Fly1. Sides: coco trunk, distance of the kid from the bottom of the coco trunk, The figure shows two pictures of a kid swinging away from the coco trunk while holding on a stalk of coco length of the coco leaf stalk. leaf. Compare the distances of the kid from the bottom of the coco trunk in these pictures. Note that the kid’s distance from the bottom of the coco trunk is farthest when he swings at full speed.2. The inequalities that exist are the following: • The distance of the kid from the bottom of the coco trunk at different QU ?E S T I ONS 1. Name the sides of the triangle formed as the kid swings away holding on to the stalk speeds. of coco leaf. • The angle determined by the coco trunk and the coco leaf stalk at different speeds 2. An inequality exists in the two triangles shown. Describe it. 3. Compare the angles formed by the coco leaf stalk and the coco trunk at the kid’s full3. Comparison: • The distance of the kid from the bottom of the coco trunk is longer speed and low speed. when he swings at full speed and shorter when he swings at low 4. How can you justify the inequality that exists between these triangles? speed. 5. Many boys and girls in the province have great fun using coco leaf stalks as swing • The angle determined by the coco trunk and the coco leaf stalk is larger when he swings at full speed and smaller when he swings at rides. Have you tried a coco leaf swing ride? low speed 6. Aside from coco leaf swing rides, what other swing rides do you know in your area or4. I can justify them deductively using the hinge theorem and its converse. from your knowledge or experience?5. (Answers may vary) 7. If you were asked to improvise a swing ride in your community, how would you design6. Possible answer: Using vines like Tarzan, swing rides in amusement the swing ride? Explain. parks) 8. Concepts on inequalities in triangles are useful in improvising a swing ride. What are7. Possible answer: Erecting a post covered with rubber or leather and the disadvantages if a designer of a swing ride does not apply these concepts? using big rope for a swing ride 9. What are the qualities of a good improvised swing ride?8. Possible disadvantages: Height of swing towers and lengths of swings 10. What are the things you should do to attain these qualities? 11. Should all designers of tools and equipment comply with standards standards and would not be proportional and can cause accidents9. Possible answers: Efficient (Strong and Stable), Safe, Well-Built, guidelines in designing them? Why? Attractive10. Possible answers: Prepare a design to determine the specifications and raw materials; let the best workers make it; have it tested for quality.11. Yes, so that tools and equipment are efficient, long-lasting, and safe to use. 456

Answer Key to Activity 22: Activity 22 YOU ARE NOW PROMOTED AS PROOFESSOR! You are Now Promoted as PROOFessor!1. 1. Write the statements supported by the reasons on the right side of the two-column proof. Statements Reasons Given: HO ≅ EP, ∠OHP > ∠EPH 1 HO ≅ EP Given Prove: OP > EH Reflexive Property of Equality 2 HP ≅ HP Given 3 ∠OHP > ∠EPH 4 OP > EH Hinge Theorem Statements Reasons2. 1 Given Statements Reasons 2 Reflexive Property of 1 ∠1 ≅ ∠2 Equality Given 3 Given 2 ∆FIH is isosceles Base angles of isosceles triangles are 4 Hinge Theorem congruent. 3 FI ≅ HI Legs of isosceles triangles are congruent. 2. Make necessary markings to the congruent angles and sides as you analyze the given and the meanings behind them. Write the reasons for the statements in the two-column 4 I is the midpoint of AT Given proof. 5 AI ≅ TI Definition of a Midpoint 6 ∠3 > ∠4 Given Given: I is the midpoint of AT, ∠1 ≅ ∠2, ∠3 > ∠4 7 HT > FA Hinge Theorem Prove: HT > FA3. Reasons Statements Reasons 1 ∠1 ≅ ∠2 1 ∠VAE ≅ ∠VEA Given 2 ∆FIH is isosceles 2 ∆AVE is an isosceles Base angles of isosceles triangles are 3 FI ≅ HI triangle. congruent 4 I is the midpoint of AT 5 AI ≅ TI 3 AV ≅ EV Legs of isosceles triangles are 6 ∠3 > ∠4 congruent. 7 HT > FA 4 FV ≅ FV Reflexive Property 5 AF > EF Given 6 ∠AVF ≅ ∠EVF Converse of Hinge Theorem 457

In this section, the discussion focuses mainly on using the triangle inequality 3. Write the statement or reason in the two-column proof. theorems in solving both real-life problems and problems that require writing proofs. Considering the application and proof-writing problems found in this module, share Given: ∠VAE ≅ ∠VEA, AF > EF your insights on the following questions: Prove: ∠AVF ≅ ∠EVF • Can you solve these problems without accurate illustrations and markings on the Reasons triangles? 1 ∠VAE ≅ ∠VEA • Can you solve these problems without prior knowledge related to triangles and writing proofs? 2 ∆AVE is an isosceles triangle. • Has your knowledge in algebra helped you in solving the problems? • Have the theorems on triangle inequalities helped you in writing proofs of 3 Legs of isosceles triangles are congruent. theorems? 4 FV ≅ FV Having tackled all concepts and skills to be learned on inequalities in triangles, revisit your decisions in Activity No.1 and write your responses to the statements under 5 Given My Decisions Later. Are there changes to your responses? Explain. 6 ∠AVF ≅ ∠EVF What would be your reply to the essential question “how can you justify inequalities In this section, the discussion focuses mainly on using the triangle inequality in triangles”? theorems in solving both real-life problems and problems that require writing proofs. Now that you have a deeper understanding of the topic, it is high time for you to put Considering the application and proof-writing problems found in this module, share your knowledge and skills to practice before you do the tasks in the next section. your insights on the following questions:Be sure to discuss the answers to the questions at the END of WHAT TO • Can you solve these problems without accurate illustrations and markings on theREFLECT AND UNDERSTAND. triangles?At this point, the students should be able to answers all the questions in • Can you solve these problems without prior knowledge related to triangles andActivity Nos. 1, 2, & 3. writing proofs?They should be able to answer the essential question “How can you justify • Has your knowledge in algebra helped you in solving the problems?inequalities in triangles?” The answer should be: Inequalities in triangles can • Have the theorems on triangle inequalities helped you in writing proofs ofbe justified deductively. When asked how, they are expected to point out thetheorems on inequalities in triangles. theorems?WWhhaatt ttooTTrraannssffeerr Having tackled all concepts and skills to be learned on inequalities in triangles, revisit your decisions in Activity No.1 and write your responses to the statements under Your goal in this section is to apply your learning to real life situations. My Decisions Later. Are there changes to your responses? Explain. You will be given a practical task which will enable you to demonstrate your understanding of inequalities in triangles. What would be your reply to the essential question “how can you justify inequalities in triangles”? Now that you have a deeper understanding of the topic, it is high time for you to put your knowledge and skills to practice before you do the tasks in the next section.458

You have to explain to the students that concepts and skills WWhhaatt ttooTTrraannssffeerrlearned in inequalities and triangles become meaningful onlywhen they can transfer their learning to real life situations such Your goal in this section is to apply your learning to real life situations. You willas performing a task where they would be able to produce be given a practical task which will enable you to demonstrate your understanding ofsomething. inequalities in triangles.Discuss to them the details of Activity No. 23 before letting them Activity 23 DISASTER PREPAREDNESS: MAKING ITstart. Give students adequate time to plan and create their THROUGH THE RAINoutputs before allowing them to present their work in class. Goal: to design and create a miniature model of a folding ladderAfter presentations, clinch the lesson by letting them share Role: a design engineerinsights on the activity questions. Audience: company head Answer Key to Activity 23:Disaster Preparedness: Making It through the Rain1. Possible Answers: Meaningful, Challenging, Fun, Interesting, Difficult Situation: The lessons learned from the widespread flooding in many parts of the country3. Possible Answers: Good planning makes the work faster; Cooperation during typhoons and monsoon season include securing tools and gadgets needed for safety. More and more people are shopping for ladders that could reach as and collaboration make the task easier and lighter; Mathematics is high as 10 feet, long enough for people to gain access to their ceiling or their roof. important in performing real-life tasks. There is a high demand for folding ladders for they can be stored conveniently.3. Yes because the concepts of hinge theorem and its converse is used in Being the design engineer of your company, your boss asks you to submit a designing the range of distances between the feet of the folding ladder miniature model of that ladder and justify the design. to make it stable enough for climbing.4. The task makes me realize that mathematics is indeed important in Product: design of a folding ladder that can reach up to 10-feet height and its miniature performing real life tasks and in creating real life tools and equipment.5. Note: Refer to the Product Column of Assessment Map for the answers Standards: accurate, creative, efficient, and well-explained/well-justified6. Example: Constructing an A-Frame House on different sizes of lot. 459

Teacher’s Note and Reminders RUBRIC Don’t CRITERIA Outstanding Satisfactory Developing Beginning RATING Forget! 43 21 Accuracy The The The The computations computations computations computations are accurate are accurate are are and show a and show erroneous erroneous wise use of the use of and show and do not the geometric geometric some use of show the concepts concepts the concepts use of the specifically specifically on triangle concepts on triangle on triangle inequalities. on triangle inequalities. inequalities. inequalities. The overall The overall The overall The overall impact of the impact of the presentation presentation impact of the impact of the of highly is impressive impressive and the use of presentation presentation and the use technology is of technology commendable. is fair and is poor and is highly Creativity commendable. the use of the use of technology is technology is evident. non-existent. Efficiency The miniature The miniature The The is very effective is effective and miniature miniature and flawlessly flawless. has some has many done. It is also defects. defects. attractive. Justification is Justification Justification Justification is not so is logically clear, is clear and clear. Some ambiguous. ideas are not Only few convincing, and convincingly connected concepts to each on triangles professionally delivered. other. Not inequalities all concepts are applied. delivered. Appropriate on triangle inequalities The concepts concepts are applied. Mathematical learned learned Justification on triangle on triangle inequalities inequalities are are applied applied. and previously learned concepts are connected to the new ones. 460

Answer Key to Activity 24: QU ?E S T I ONS 1. How do you find the experience of designing? Final Construction of Concept Museum 2. What insights can you share from the experience? 3. Has the activity helped you justify inequalities in triangles? How?Please refer to answer key of Activity 3, Hello, Dear Concept Contractor. 4. How did the task help you see the real world use of the concepts on Teacher’s Note and Reminders inequalities in triangles? 5. Aside from designing a folding ladder, list down the real-life Don’t Forget! applications of concepts learned in Inequalities in Triangles from this module. 6. Can you think of other real-life applications of this topic? SUMMARY Activity 24 FINAL CONSTRUCTION OF CONCEPT MUSEUM Directions: After learning all the concepts and skills on Inequalities in Triangles, take a final visit to your responses in Activity No.3— Hello, Dear Concept Contractor—of this module and make some modifications of or corrections to your responses and their corresponding justifications. TH E Write two Knowing TH>TX>HX, Write three inequalities to Write two Inequalities to what question involving describe the sides of Inequalities to describe angle 1. this triangle describe angle 2. inequality should you use to check if they form a triangle? MY 1 X CONCEPT N 2 M Write the combined MUSEUM 3 Write an if-then 4 C inequality you will on TRIANGLE statement about use to determine INEQUALITIES the length of Come visit now! the sides given the MK? marked angles K Write if-then B6 Write if-then 5 7R Write a detailed if- statement about the Write statement about then statement to angles given the an the sides given Write a detailed describe triangles marked sides. if-then statement to MXK and KBF if if-then the marked describe triangles angle X is larger the angles. than angle B statement about MXK and KBF if MK is longer angles given the than KF. marked sides. FW 461

Answer Key to Activity 24: Activity 25 CONCEPTS I’VE LEARNED THAT LAST Final Construction of Concept Museum FOREVERPlease refer to answer key of Activity 3, Hello, Dear Concept Contractor. Direction: Fill in the blanks with the right words to make the statements complete. I have learned that inequalities in triangles, even without actual measurements, can be justified (in this manner) deductively usingtheorems on inequalities in triangles.I’ve learned that concepts on inequalities in triangles are usefulto air traffic controllers to avoid in designing devices and tools thatcollision among in designing make use of hinges likeaircrafts in busy the layout of the folding ladder. airports. refrigerator, sink, and stove of a kitchen.in designing in designing rides in in exercising roofs amusement parks like my skills in writing direct and indirectof houses. swing rides proof.You have completed the lesson on Inequalities in Triangles. Before you go to the You have completed the lesson on Inequalities in Triangles. Before you go to thenext geometry lesson on Parallelism and Perpendicularity, you have to answer a next geometry lesson on Parallelism and Perpendicularity, you have to answer apost-assessment and a summative test. post-assessment and a summative test.Before giving the post-assessment, let the groups to review their answersto the items in pre-assessment and make corrections. Let them explain theirletters of choices. You may ask them to explain why other choices are wrong. 462

POST-ASSESSMENT: Let’s find out how much you already learn about this topic. On a separate sheet, write only the letter of the choice that youthink best answers the question. Please answer all items.1. Which of the following is not an inequality theorem for one triangle? a. Triangle Inequality Theorem 1 (SsAa) b. Triangle Inequality Theorem 3 (S1 + S2 > S3) c. Exterior Angle Inequality Theorem d. Hinge Theorem2. Which of the following angles is an exterior angle of ∆RPY? UT R ∠7 71 2 P ∠4 d. 5 43 6 Y a. ∠2 b. ∠3 c. 3. Study the figure in no. 2. Notice that m∠5 > m∠3 and m∠5 > m∠1. Which theorem justifies these observations? a. Triangle Inequality Theorem 1 (SsAa) b. Triangle Inequality Theorem 2 (AaSs) c. Triangle Inequality Theorem 3 (S1 + S2 > S3) d. Exterior Angle Inequality Theorem 463

4. Chris forms triangles by bending a 16-inch wire. Which of the following sets of wire lengths successfully form a triangle? I. 4 in, 5 in, 6 in III. 4 in, 5 in, 7 in II. 4 in, 4 in, 8 in IV. 3 in, 4 in, 9 in a. I, II b. III, IV c. II, IV d. I, III5. From the inequalities in the triangles shown, Jarold concluded that ∠OHM > ∠EHM. Which theorem on inequalities in triangle justifies his answer? O 10 8 HM 10 7 E a. Triangle Inequality Theorem 3 ((SSs1+ AS2a)> S3) b. Triangle Inequality Theorem 1 c. Converse of Hinge Theorem d. Hinge Theorem6. Kyle has proved that IS > IW. Which of the following statements is NOT part of his proof? W 10 E 75o 10 S I a. ES ≅ EW c. EI ≅ EI b. ∠WEI + ∠SEI = 180 d. ∠W < ∠S 464

7. What theorem should Kyle use to justify his proved statement in no. 5? a. Hinge Theorem b. Converse of Hinge Theorem c. Triangle Inequality Theorem 1 (SsAa) d. Triangle Inequality Theorem 3 (S1 + S2 > S3)8. Chloe studies the triangles in the figure carefully. Which should be her final conclusion? T 8 9I E 5 5 M a. TM ≅ TM c. IM ≅ EM b. ET > IT d. ∠EMT > ∠ITM9. Which theorem justifies Chloe’s conclusion in no. 8? a. Hinge Theorem b. Converse of Hinge Theorem c. Triangle Inequality Theorem 1 (SsAa) d. Triangle Inequality Theorem 3 (S1 + S2 > S3)10. In ∆GUD, GU = DU and GD > DU. Which of the following statements may NOT be true? a. GU < GD − DU b. m∠U > m∠D C. m∠U > m∠G D. m∠D = m∠G 465

11. In ∆TRY, if TR = 3, RY = 5, and TY = 2, which statement is true? a. m∠R > m∠Y c. m∠Y > m∠T b. m∠R > m∠T d. m∠T > m∠R12. Which theorem justifies the then-statement in no. 11? a. Triangle Inequality Theorem 1 (SsAa) b. Triangle Inequality Theorem 2 (AaSs) c. Triangle Inequality Theorem 3 (S1 + S2 > S3) d. Exterior Angle Inequality Theorem13. From a rendezvous, 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 right turns at different angles—Oliver at an angle of 30o and Ruel at 40o. Both continue hiking and cover another four kilometers each before taking a rest. To find out who is farther from the rendezvous, select the illustration that describes appropriately the problem. a. c. b. d. 466

14. Which theorem of inequality in triangles helps you in determining who is farther from the rendezvous? A. Hinge Theorem B. Converse of Hinge Theorem c. Triangle Inequality Theorem 1 (SsAa) d. Triangle Inequality Theorem 3 (S1 + S2 > S3)For items no. 15-20, use the situation described.Your friend asks for your suggestion on how to raise the height of his tent without changing the amount of area it covers. Original Tent15. Which of the following designs meet the qualifications of your friend? I II III IV a. I and III b. II and III c. III and IV d. II, III, and IV 467

16. Which design/s is/are contradictory to your friend’s specifications? a. I only b. IV only c. I and II d. I and IV17. Which design requires more tent material? a. I b. II c. III d. IV18. The modified tents have equal heights. Which design is the most practical and easiest to assemble? a. I b. II c. III d. IV19. What theorem of inequality in triangles justifies design no. IV? a. Triangle Inequality Theorem 1 (SsAa) b. Triangle Inequality Theorem 2 (AaSs) c. Triangle Inequality Theorem 3 (S1 + S2 > S3) d. Exterior Angle Inequality Theorem20. Which insights have you learned from the tent designs? I. The steeper the roof of a tent, the less area it covers. II. The larger the roof angle of a tent, the wider the area it covers. III. Modifying a tent design does not always require money. a. III only b. I, II c. I, III d. I, II, IIIAnswer Key to Post-Assessment: 1. D 6. B 11. D 16. A 17. C2. A 7. A 12. A 18. D 19. C3. D 8. D 13. D 20. D4. D 9. B 14. A5. C 10. A 15. D 468

TEACHING GUIDEModule 9: Parallelism and PerpendicularityA. Learning Outcomes Content Standard: The learner demonstrates understanding of the key concepts of parallel and perpendicular lines. Performance Standard: The learner is able to communicate mathematical thinking with coherence and clarity in solving real-life problems involving parallelism and perpendicularity using appropriate and accurate representations.SUBJECT: LEARNING COMPETENCIESGrade 8 Illustrates parallel and perpendicular lines.Mathematics * Illustrates and proves properties of parallel lines cut by a transversal. * Determines and proves the conditions under which lines and segments are parallel or perpendicular.QUARTER: * Determines the conditions that make a quadrilateral a parallelogram and prove that a quadrilateral is aThird Quarter parallelogram.STRAND: * Uses properties to find measures of angles, sides, and other quantities involving parallelograms.Geometry ESSENTIAL UNDERSTANDING: ESSENTIAL QUESTION:TOPIC:Parallellism and Students will understand that: How can parallelism or perpendicularity ofPerpendicularity lines be established?LESSON: The establishment of parallelism and perpendicularity of lines in • Are you sure that the given lines areParallellism and real life may be done through deductive reasoning. parallel?Perpendicularity • Lines can be related in different ways such as parallel, • Are you sure that the given lines are perpendicular, intersecting and skew. perpendicular? • Concepts of parallel and perpendicular lines can be determined deductively. When two parallel lines are cut by a transversal, pairs of angles can be related as either congruent or supplementary. TRANSFER GOAL: Students will on their own use the key concepts of parallelism and perpendicularity of lines in solving real-life problems. 469

B. Planning for AssessmentProduct/PerformanceThe following are products and performances that students are expected to accomplish after with in this module.a. Active participation in the different activities presented in the module will be evident.b. Real-life problems involving parallelism and perpendicularity of lines will be solved.c. Proofs are completed and devised.d. A model of a book case that displays students’ understanding on the concepts of parallelism and perpendicularity of lines will be created. Assessment Map KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PRODUCT TYPE Pre–Test Pre–Test Pre–Test Pre–Test Items Nos. 15-20 Pre-Assessment/ Items Nos. 1-4 Items Nos. 5-8 Items Nos. 9-14 Diagnostic Design Analysis Generalization Table (Application) Formative Assessment (Self-Knowledge) A-R Guide (Self-Knowledge) Worksheet in Writing Proofs Activity 14 (Explanation, Application, Interpretation) Open-Ended Exercises in Proving Activities 15 & 16 (Explanation, Application, Interpretation) Generalization Table (Explanation, Interpretation, Self-Knowledge) Think Twice! Draw Me Right! (Application) Problem Solving Worksheets Activities 6 & 13 (Application, Interpretation) 470

Summative Assessment Generalization Table Transfer Task Self-Assessment (Explanation, Interpretation, Self-Knowledge) (Application, Empathy, A-R Guide (Revisit) Perspective) (Interpretation, Self-Knowledge) Concept Mapping (Explanation, Interpretation) Summative Test (Application, Interpretation, Explanation) Post Test (Application, Interpretation) Lesson Closure (Explanation, Self- Knowledge) Assessment Matrix Levels of What will I assess? How will I assess? How Will I Score?Assessment Every correct answer is given 1 point. Knowledge * Illustrates parallel and perpendicular Paper and Pencil Test Every correct answer with correct solution is 15% lines. Activity 22 given 2 points. (Summative Test, Part A)Process/Skills 25% * Illustrates and proves properties of Every correct answer is given 1 point. parallel lines cut by a transversal. Post Test Refer to the Rubric in Writing Proof. (Item Nos. 1-3) Every correct answer is given 1 point. * Determines and proves the conditions under which lines and segments are Paper and Pencil Test parallel or perpendicular. Activity 22 (Summative Test, Part B) * Determines the conditions that make a quadrilateral a parallelogram and prove Post Test that a quadrilateral is a parallelogram. (Item Nos. 4-8) * Uses properties to find measures of angles, sides, and other quantities involving parallelogramsUnderstanding Paper and Pencil Test 30% Activity 22 (Summative Test, Part C) Post Test (Item Nos. 9-14) 471

* uses the key concepts of parallelism and Transfer Task perpendicularity of lines in solving real- Activity 23 Refer to the Rubric for Performance Task Every correct answer is given 1 point.Product life problems. (Designer’s Forum) 30% Post Test (Item Nos. 15-20)C. Planning for Teaching-LearningIntroduction: The module is all about Parallelism and Perpendicularity. It gives emphasis on the theorems involving parallel andperpendicular lines, proving properties of parallel lines cut by a transversal, the conditions to prove that a quadrilateral is aparallelogram and applications of parallelism and perpendicularity. The students are given various activities that will enablethem to set up parallelism and perpendicularity and use the important concepts in solving real-life problems. These activitiescan be given in the form of a game or a worksheet to be done inside the classroom or these can be given as their assignment.These can also be done individually or collaboratively. Activities may or may not be graded depending on the teacher’sdiscretion but all results will be recorded and kept for evaluation purposes. Some brain teasers are also presented in thisguide and these can be used to catch students’ attention. As an introduction to the main lesson, ask the students the following questions: Have you ever wondered how carpenters, architects and engineers design their work? What factors are being considered in making their designs? The use of parallelism and perpendicularity of lines in real life necessitates the establishment of these conceptsdeductively. This module seeks to answer the question: “How can we establish parallelism or perpendicularity of lines”? Let the students take the pre-assessment to find out how much they already know about the module. Instruct them to choosethe letter that corresponds to the best answer and write it on a separate sheet. After taking the test, let them note the items thatthey were not able to answer correctly. 472

PRE - ASSESSMENT Find out how much you already know about this module. Choose the letter that corresponds to the best answer and write iton a separate sheet. Please answer all items. After taking this short test, take note of the items that you were not able to answercorrectly. Correct answers are provided as you go through the module.(K)1. Using the figure below, if l1 || l2 and t is a transversal, then which of the following are corresponding angles? a. ∠4 and ∠6, ∠3 and ∠5 t b. ∠1 and ∠7, ∠2 and ∠8 l1 12 43 c. ∠1 and ∠5, ∠2 and ∠6 l2 56 d. ∠4 and ∠5, ∠3 and ∠6 87Answer: c, Corresponding angles are pair of non-adjacent angles on the same side of the transversal, one interiorand one exterior.(K)2. All of the following are properties of a parallelogram except: a. Diagonals bisect each other. b. Opposite angles are congruent. c. Opposite sides are congruent. d. Opposite sides are not parallel. Answer: d, A quadrilateral is a parallelogram if two pairs of opposite sides are parallel and congruent.(K)3. Lines m and n are parallel cut by transversal t which is also perpendicular to m and n. Which statement is not correct? mn a. ∠1 and ∠6 are congruent. b. ∠2 and ∠3 are supplementary. c. ∠3 and ∠5 are congruent angles. d. ∠1 and ∠4 form a linear pair. t 12 34 56 78 Answer: d, Linear pairs are adjacent angles. 473

(K)4. Using the figure below, which of the following guarantees that m || n? ∠1 ≅ ∠7 t ∠3 ≅ ∠5 a. ∠4 ≅ ∠5 n 12 b. ∠4 ≅ ∠7 34 c. d. m 56 78 Answer: c, If two parallel lines are cut by a transversal, alternate interior angles are congruent.(S)5. Lines a and b are parallel cut bmy transversal m. If m∠1 = 85, what is the measure of ∠5? a. 80 a 21 b. 85 34 c. 95 d. 100 b 87 56 Answer: b, ∠1 and ∠5 are alternate-exterior angles and are congruent.(S)6. JOSH is a parallelogram, m∠J = 57, find the measure of ∠H. a. 43 b. 57 c. 63 d. 123 Answer: b, ∠J and ∠H are consecutive angles of the parallelogram, therefore, they are supplementary.(S)7. Using the figure below, if m || n and t is a transversal, which angles are congruent to ∠5? t a. ∠1, ∠2 and ∠3 n 12 b. ∠1, ∠4 and ∠8 34 c. ∠1, ∠4 and ∠7 m 56 d. ∠1, ∠2 and ∠8 78 Answer: b, ∠5 and ∠1 are corresponding angles thus ∠5 ≅ ∠1 and then ∠1 and ∠4 are vertical angles so ∠1 ≅ ∠4 ≅ ∠5. Lastly, ∠5 and ∠8 are vertical angles therefore ∠5 ≅ ∠8. 474

(S)8. LOVE is a parallelogram. If SE = 6, then what is SO? LO a. 3 S b. 6 c. 12 d. 15 E VAnswer: b, Diagonals of a parallelogram bisect each other. Thus, SE ≅ SO and the measures would be the same orequal.(U)9. The Venn Diagram below shows the relationships of quadrilaterals. Which statements are true? I - Squares are rectangles. Quadrilaterals II- A trapezoid is a parallelogram. Parallelograms III- A rhombus is a square. IV- Some parallelograms are squares. Rectangle Square Rhombus a. I and II Trapezoid b. III and IV c. I and IV d. II and IIIAnswer: c, Only I and IV are true statements based on the properties of parallelograms(U)10. All of the figures below illustrate parallel lines except: a. c. b. d. Answer: d, Lines that are not coplanar would never be parallel. 475

(U)11. In the figure below, a ║ d with e as the transversal. What must be true about ∠3 and ∠4 if b ║ c with e, also as the transversal? e a b a. ∠3 is a complement of ∠4. b. ∠3 is congruent to ∠4. 14 c. ∠3 is a supplement of ∠4. 32 d. ∠3 is greater than ∠4. c dAnswer: b, To make b║c ∠3 should be congruent to ∠4, since alternate interior angles of parallel lines cut by atransversal are congruent.(U)12. Which of the following statements ensures that a quadrilateral is a parallelogram? a. Diagonals bisect each other. b. The two diagonals are congruent. c. The consecutive sides are congruent. d. Two consecutive angles are congruent. Answer: a, The other statements don’t guarantee that the quadrilateral is a parallelogram. An isosceles trapezoid has congruent diagonals. A trapezoid can also have pairs of consecutive sides that are congruent as well as consecutive angles.(U) 13. Which of the following statements is always true? a. Lines that do not intersect are parallel lines. b. Two coplanar lines that do not intersect are parallel lines. c. Lines that form a right angle are parallel lines. d. Skew lines are parallel lines. Answer: b, Parallel lines are coplanar lines that do not intersect. 476

(U)14. STAR is a rhombus with diagonal RT, if m∠STR = 3x – 5 and m∠ART = x + 21. What is m∠RAT? S T a. 13 b. 34 c. 68 d. 112 R A STAR is a rhombus, therefore ST is parallel to RA and RT is a transversal. ∠STR and ∠ART are Answer: d, congruent because they are alternate interior angles thus x = 13. Since a diagonal of a rhombus bisects opposite angles, the measure of ∠STA and ∠ARS is 34x2 or 68 each. ∠SRA and ∠RAT are consecutive angles and so they are supplementary, so m∠RAT is 112.(P)15. You are tasked to divide a blank card into three equal rows/pieces but you do not have a ruler. Instead, you will use a piece of equally lined paper and a straight edge. What is the sequence of the steps you are going to undertake in order to apply the theorem on parallel lines? I – Mark the points where the second and third lines intersect the card. II – Place a corner of the top edge of the card on the first line of the paper. III – Repeat for the other side of the card and connect the marks. IV – Place the corner of the bottom edge on the fourth line. a. I, II, III, IV b. II, III, IV, I c. I, III, IV, II d. II, IV, I, III Answer: d, Since the lines of the paper are equally spaced, the lines are parallel.(P)16. You are a student council president. You want to request for financial assistance for the installation of a book shelf for the improvement of your school’s library. Your student council moderator asked you to submit a proposal for their approval. Which of the following will you prepare to ensure that your request will be granted? I. design proposal of the book shelf II. research on the importance of book shelf 477

III. estimated cost of the project IV. pictures of the different libraries a. I only b. I and II only c. I and III only d. II and IV only Answer: c, The cost of the project is one of the important thing to consider and of course the cost is dependent upon the proposed design of the book shelf.(P)17. Based on your answer in item no. 16, which of the following standards should be the basis of your moderator in approving or granting your request? a. accuracy, creativity and mathematical reasoning b. practicality, creativity and cost c. accuracy, originality and mathematical reasoning d. organization, mathematical reasoning and cost Answer: b, Since financial aspect is involved thus there is a need to consider cost of the project. But despite of its cost it should still be creative and feasible or realistic.(P)18. Based on item no. 16, design is common to all the four given options. If you were to make the design, which of the illustrations below will you make to ensure stability? a. b. c. d. Answer: a, The key concepts about parallelism will ensure its stability. 478

(P)19. You are an architect of the design department of a mall. Considering the increasing number of mall-goers, the management decided to restructure their parking lot so as to maximize the use of the space. As the head architect, you are tasked to make a design of the parking area and this design is to be presented to the mall administrators for approval. Which of the following are you going to make so as to maximize the use of the available lot? a. c. b. d. Answer: b, Vertical parallel parking will surely maximize the available lot. (P)20. Based on your answer in item no. 19, how will your immediate supervisor know that you have a good design? a. The design should be realistic. b. The design should be creative and accurate. c. The design should be accurate and practical. d. The design shows a depth application of mathematical reasoning and it is practical. Answer: d, A good design should apply mathematical reasoning aside from its practicality. 479

WWhhaatt ttoo KKnnooww LEARNING GOALS AND TARGET: The module starts with the hook activity on Optical Illusions where students are encouraged to answer certain questions. Students may be deceived by • The learner demonstrates understanding of the key concepts of parallel and the pictures when presented in larger scale. This can be done as a class perpendicular lines. activity. Encourage students to explain their answers. • The learner is able to communicate mathematical thinking with coherence and Teacher’s Note and Reminders clarity in solving real-life problems involving parallelism and perpendicularity using appropriate and accurate representations. WWhhaatt ttoo KKnnooww Start the module by taking a look at the figures below and then answer the succeeding questions. Activity 1 OPTICAL ILLUSION Don’t • Can you see straight lines in the pictures above? ___________________Forget! • Do these lines meet/intersect? ___________________ • Are these lines parallel? Why? ___________________ • Are the segments on the faces of the prism below parallel? Why?_________ • Can you describe what parallel lines are? ___________________ • What can you say about the edges of the prism? ___________________ • Are these lines perpendicular? Why? ___________________ • Can you describe what perpendicular lines are? ___________________ 480

Students have just tried describing parallel and perpendicular lines. In their You have just tried describing parallel and perpendicular lines. In Activities 2 andnext activities, their prior knowledge on parallelism and perpendicularity will 3, your prior knowledge on parallelism and perpendicularity will be extracted.be elicited.They will be given individual activities such as Generalization Table and Activity 2 GENERALIZATION TABLEAnticipation-Reaction Guide.Copy of each activity can be reproduced or be copied on a separate notebook Direction: Fill in the first column of the generalization table below by stating your initialwhere they can write their answers. thoughts on the question.Keep the answer sheets of the students in Activities 2 and 3 for future use. “How can parallelism or perpendicularity of lines be established?” Teacher’s Note and Reminders My Initial Thoughts Activity 3 AGREE OR DISAGREE! ANTICIPATION-REACTION GUIDE Read each statement under the column TOPIC and write A if you agree with the statement; otherwise, write D. Don’t Before-Lesson TOPIC: Parallelism and PerpendicularityForget! Response 1. Lines that do not intersect are parallel lines. 2. Skew lines are coplanar. 3. Transversal is a line that intersects two or more lines. 4. Perpendicular lines are intersecting lines. 5. If two lines are parallel to a third line, then the two lines are parallel. 481

Instead of presenting Activity 4 as a worksheet, the teacher may present 6. If two lines are perpendicular to the same line,it as an outdoor activity where the figure will be drawn on the ground and then the two lines are parallel.pairs of students will step on the pairs of angles mentioned by the teacher.For classes that are technology-ready, they may use the link for the activity. 7. If one side of a quadrilateral is congruent tohttp://www.mathwarehouse.com/geometry/angle/transveral-and-angles.php its opposite side, then the quadrilateral is a parallelogram. Teacher’s Note and Reminders 8. Diagonals of a parallelogram bisect each other. 9. Diagonals of a parallelogram are congruent. 10. Diagonals of a parallelogram are perpendicular. 11. Opposite sides of a parallelogram are parallel. 12. Opposite angles of a parallelogram are congruent. 13. Consecutive angles of a parallelogram are congruent. 14. Squares are rectangles. 15. Squares are rhombi. Activity 4 NAME IT! A RECALL... We see parallel lines everywhere. Lines on a pad paper, railways, edges of a door or window, fence, etc. suggest parallel lines. Complete the table below using the given figure as your reference: p 12 m 34 Don’t n 56Forget! 78 Corresponding Alternate Alternate Same Side Same Side Angles Interior Angles Exterior Angles Interior Angles Exterior Angles 482

WWhhaatt ttoo PPrroocceessss You gave your initial ideas on naming angle pairs formed by two lines cut by a transversal. What you will learn in the next sections will enable you to do the final Your goal in this section is to let the students learn, and understand project which involves integrating the key concepts of parallelism and perpendicularity the key concepts of measurement of angles formed by parallel lines cut by of lines in model making of a book case. Now find out how these pairs of angles a transversal and basic concepts about perpendicularity and the properties are related in terms of their measures by doing the first activity on investigating the of parallelogram. Towards the end of this section, the students will be relationship between the angles formed by parallel lines cut by a transversal. encouraged to learn the different ways of proving. WWhhaatt ttoo PPrroocceessss Students will be given activities such as investigation, worksheets and some formative assessments where the decision of whether they will be Your goal in this section is to learn and understand key concepts of measurement graded or not will depend on the teacher. of angles formed by parallel lines cut by a transversal and basic concepts about Discussions on parallelism, perpendicularity, properties of parallelogram, perpendicularity and the properties of parallelogram. Towards the end of this section, and writing proofs will also be done in this section. Assessments will also you will be encouraged to learn the different ways of proving deductively. You may follow after the discussion. also visit the link for this investigation activity. http://www.mathwarehouse.com/ The generalization table will again be presented in order to keep track geometry/angle/interactive-transveral-angles.php and evaluate students’ understanding about the lesson. Make sure to keep their answer sheets or let the students keep theirs. Activity 5 LET’S INVESTIGATE! Two parallel lines when cut by a transversal form eight angles. This activity will lead Activity 5 needs advance preparation for the teacher. Reproduce the activity you to investigate the relationship between and among angles formed. sheet and assign the students to bring their own protractors. This activity Measure the eight angles using your protractor and list all inferences or observations can be done in pairs. Students are expected that they already know how to in the activity. measure an angle. If possible, students may also visit the link. http://www. mathwarehouse.com/geometry/angle/interactive-transveral-angles.php m∠1= ________ 1 2 m∠2= ________ Teacher’s Note and Reminders 3 4 m∠3= ________ m∠4= ________ Don’t 75 86 mm∠∠65== Forget! ________ ________ m∠7= ________ m∠8= ________ OBSERVATIONS: ______________________________________________________________________ ______________________________________________________________________ _________________________________________________________. Now, think about the answers to the following questions. Write your answers in your answer sheet. 483

After doing the investigation, encourage the students to answer the questions. QU ?E S T I ONS 1. What pairs of angles are formed when two lines are cut by aThis can be done orally or they may continue working in pairs. transversal line?Students can list any pair of angles and then classify which pairs have equal 2. What pairs of angles have equal measures? What pairs of anglesmeasures and which are supplementary. are supplementary?Students can list any pair of angles and then classify which pairs have equal 3. Can the measures of any pair of angles (supplementary or equal)measures and which are supplementary. guarantee the parallelism of lines? Support your answer.At this point, discuss important concepts about parallelism and 4. How can the key concepts of parallel lines facilitate solving real-perpendicularity of lines. Always keep on asking their ideas first from time to life problems using deductive reasoning?time and make sure to clarify/explain the misconceptions. For example, NOTALL LINES THAT DO NOT INTERSECT ARE PARALLEL LINES. That is Discussion: Parallelismbecause we have the SKEW LINES. These lines are non-coplanar lines thatdo not intersect. Another misconception is on transversal. A TRANSVERSAL 1. Two lines are parallel if and only if they are coplanar and they do not intersect.IS NOT JUST A LINE THAT INTERSECTS TWO OR MORE LINES. It should tintersect those lines at DIFFERENT POINTS. (m || n) Teacher’s Note and Reminders m 12 34 n 75 86 transversal 2. A line that intersects two or more lines at different points is called a transversal. a. The angles formed by the transversal with the two other lines are called: • exterior angles (∠1, ∠2, ∠7 and ∠8) • interior angles (∠3, ∠4, ∠5 and ∠6). b. The pairs of angles formed by the transversal with the other two lines are called: • corresponding angles (∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8) • alternate-interior angles (∠3 and ∠6, ∠4 and ∠5) • alternate-exterior angles (∠1 and ∠8, ∠2 and ∠7) • interior angles on the same side of the transversal (∠3 and ∠5, ∠4 and ∠6) • exterior angles on the same side of the transversal (∠1 and ∠7, ∠2 and ∠8) Don’t 3. If two lines are cut by a transversal, then the two lines are parallel if:Forget! a. corresponding angles are congruent. b. alternate-interior angles are congruent. c. alternate-exterior angles are congruent. d. interior angles on the same side of the transversal are supplementary. e. exterior angles on the same side of the transversal are supplementary. To strengthen your knowledge regarding the different angles formed by parallel lines cut by a transversal line and how they are related with one another, you may visit the following sites: http://www.youtube.com/watch?v=AE3Pqhlvqw0&feature=related http://www.youtube.com/watch?v=VA92EWf9SRI&feature=relmfu 484

To check their understanding in the previous discussion, encourage them to Activity 6 UNCOVERING THE MYSTERY OF PARALLELanswer Activity 6. LINES CUT BY A TRANSVERSAL Answer Key Study the problem situation below and answer the succeeding questions:Activity 6 A zip line is a rope or a cable that you can ride down on a pulley. The pair of1. Each pair of angles is supplementary. They are interior angles on the zip lines below goes from a 20- foot tall tower to a 15- foot tower 150 meters away same side of the transversals MA and HT.2. Equate each pair of angles to 1800. This will result to z = 330 and y = 820. in a slightly inclined ground as shown in the sketch. (Note: Tension of the rope is m∠M = 81 m∠A = 99 excluded.) 3z A m∠MHT = 98 b m∠ATH= 82 M3. Yes, because pairs of interior angles on the same side of each transversal 2z +15 are supplementary.4. No, because pairs of interior angles on the same side of each transversal a (tower) are not supplementary. y + 18 yTTo strengthen their discoveries regarding the different angles formed by Hparallel lines cut by a transversal and how they are related with one another,they may visit the following sites: 1. What kind of angle pairs are ∠M and ∠A? ∠MHT and ∠ATH?http://www.youtube.com/watch?v=AE3Pqhlvqw0&feature=related _____________________________________________________http://www.youtube.com/watch?v=VA92EWf9SRI&feature=relmfu _____________________________________________________For practice, students may proceed to this link: 2. Using the given information stated in the figure, what are the measures of thehttp://www.regentsprep.org/Regents/math/geometry/GP8/PracParallel.htm four angles? Solution: Answers: m∠M = _________ m∠A = _________ m∠MHT = _________ m∠ATH = _________ 3. Are the two towers parallel? Why do you say so? 4. Is the zip line parallel to the ground? Why do you say so? For practice you may proceed to this link: http://www.regentsprep.org/Regents/math/geometry/GP8/PracParallel.htm 485

Activity 7 can given as an individual activity using paper and pen. Activity 7 LINES AND ANGLESAnswers to Part III1. x = 500 2. x = 550 3. x = 340 I. Study the figure and answer the following questions as accurate as you can. The figure below shows a || b with t as transversal. Teacher’s Note and Reminders a b Don’t Forget! t 3 4 1 2 7 6 8 5 Name: _________ 1. 2 pairs of corresponding angles __________ _________ 2. 2 pairs of alternate interior angles __________ _________ 3. 2 pairs of alternate exterior angles __________ _________ 4. 2 pairs of interior angles on the same __________ side of the transversal _________ 5. 2 pairs of exterior angles on the same __________ side of the transversal II. Given m ║ n and s as transversal. s m 12 34 n 56 78 1. Name all the angles that are congruent to ∠1. _______________ 2. Name all the angles that are supplement of ∠2. _______________ III. Find the value of x given that l1 ║ l2. l1 l2 1. m∠1 = 2x + 25 and m∠8 = x + 75 ________ 12 5 6 3 4 78 2. m∠2 = 3x – 10 and m∠6 = 2x + 45 ________ 3. m∠3 = 4v – 31 and m∠8 = 2x + 7 ________ 486

Before discussing Perpendicularity, present Activity 8. You can think of a way Activity 8 AM I PERPENDICULAR? LET’S FIND OUT….!to make it more interactive. The process questions can be done as a classactivity. Given any two distinct lines on a plane, the lines either intersect or are parallel. IfYou may suggest to the students the following link to learn more about two lines intersect, then they form four angles. Consider the figures below to answer theperpendicular lines: http://www.mathsisfun.com/geometry/parallel-perpendicular-lines-planes.html questions that follow. Teacher’s Note and Reminders n a m Figure 1 Figure 2 b s l1 t Figure 3 l2 Figure 4 QU ?E S T I ONS 1. What is common in the four figures given above? ___________________________________________________ Don’t 2. What makes figures 3 and 4 different from the first two figures?Forget! ___________________________________________________ 3. What does this symbol indicate? ___________________________________________________ 4. Which among the four figures show perpendicularity? Check by using your protractor. ___________________________________________________ 5. When are the lines said to be perpendicular to each other? ___________________________________________________ 6. How useful is the knowledge on perpendicularity in real-life? Cite an example in which perpendicularity is said to be important in real-life. ___________________________________________________ ___________________________________________________ ___________________________________________________ 487

Students may watch the video lessons using the given links. These videos will Discussion: Perpendicularityexplain how to construct a perpendicular line to a point and a perpendicularline through a point not on a line. Two lines that intersect to form right angles are said to be perpendicular. This ishttp://www.youtube.com/watch?v=dK3S78SjPDw&feature=player_ not limited to lines only. Segments and rays can also be perpendicular. A perpendicularembedded bisector of a segment is a line or a ray or another segment that is perpendicular to thehttp://www.youtube.com/watch?feature=player_embedded&v=_jWw_ segment and intersects the segment at its midpoint. The distance between two parallelnQTtCwhttp://www.youtube.com/watch?v=9ZjSz199Huc&feature=player_ lines is the perpendicular distance between one of the lines and any point on the otherembedded: line. Teacher’s Note and Reminders Perpendicular 90oPerpendicular 90o P XY Z perpendicular distance between the parallel lines Perpendicular bisector (XY ≅ YZ) The small rectangle drawn in the corner indicates “right angle”. Whereas, ⊥ is a symbol use to indicate perpendicularity of lines as in XZ ⊥ PY. To prove that two lines are perpendicular, you must show that one of the following theorems is true: 1. If two lines are perpendicular to each other, then they form four right angles. Don’t m If m ⊥ n, then we canForget! conclude that ∠1, ∠2, ∠3 12 and ∠4 are right angles. n 34 488