Mathematics 9 Quarter 3 Self-Learning Module 3 Solving for Sides and Diagonals of Parallelograms Mathematics Grade 9 Quarter 3 – Self-Learning Module 2: Solving for Sides and Diagonals of Parallelograms First Edition, 2020 Republic Act 8293, Section 176 states that no copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education - Schools Division of Pasig City Development Team of the Self-Learning Module Writer: Riza R. Noceto Editors: Ma. Cynthia P. Badana; Ma. Victoria L. Peñalosa Reviewers: Julie R. Reyes; Roberta B. Tuando; Raneth A. Yago (technical) Illustrator: Riza R. Noceto Layout Artist: Management Team: Ma. Evalou Concepcion A. Agustin OIC – Schools Division Superintendent Aurelio G. Alfonso EdD OIC - Assistant Schools Division Superintendent Victor M. Javeña EdD Chief – School Governance & Operations Division and OIC – Chief Curriculum Implementation Division Education Program Supervisors Librada L. Agon EdD (EPP/TLE/TVL/TVE) Liza A. Alvarez (Science/STEM/SSP) Bernard R. Balitao (AP/HUMSS) Joselito E. Calios (English/SPFL/GAS) Norlyn D. Conde EdD (MAPEH/SPA/SPS/HOPE/A&D/Sports) Wilma Q. Del Rosario (LRMS/ADM) Ma. Teresita E. Herrera EdD (Filipino/GAS/Piling Larangan) Perlita M. Ignacio PhD (EsP) Dulce O. Santos PhD (Kindergarten/MTB-MLE) Teresita P. Tagulao EdD (Mathematics/ABM) Printed in the Philippines by Department of Education – Schools Division of Pasig City Mathematics 9 Quarter 3 Self-Learning Module 3 Solving for Sides and Diagonals of Parallelograms Introductory Message For the Facilitator: Welcome to the Mathematics Grade 9 Self-Learning Module on Solving for Sides and Diagonals of Parallelograms This Self-Learning Module was collaboratively designed, developed and reviewed by educators from the Schools Division Office of Pasig City headed by its Officer-in-Charge Schools Division Superintendent, Ma. Evalou Concepcion A. Agustin, in partnership with the City Government of Pasig through its mayor, Honorable Victor Ma. Regis N. Sotto. The writers utilized the standards set by the K to 12 Curriculum using the Most Essential Learning Competencies (MELC) in developing this instructional resource. This learning material hopes to engage the learners in guided and independent learning activities at their own pace and time. Further, this also aims to help learners acquire the needed 21st century skills especially the 5 Cs, namely: Communication, Collaboration, Creativity, Critical Thinking, and Character while taking into consideration their needs and circumstances. In addition to the material in the main text, you will also see this box in the body of the self-learning module: Notes to the Teacher This contains helpful tips or strategies that will help you in guiding the learners. As a facilitator you are expected to orient the learners on how to use this selflearning module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Moreover, you are expected to encourage and assist the learners as they do the tasks included in the self-learning module. For the Learner: Welcome to the Mathematics Grade 9 Self-Learning Module on Solving for Sides and Diagonals of Parallelograms This self-learning module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning material while being an active learner. This self-learning module has the following parts and corresponding icons: Expectations - This points to the set of knowledge and skills that you will learn after completing the module. Pretest - This measures your prior knowledge about the lesson at hand. Recap - This part of the module provides a review of concepts and skills that you already know about a previous lesson. Lesson - This section discusses the topic in the module. Activities - This is a set of activities that you need to perform. Wrap-Up - This section application of the lesson. summarizes the concepts and Valuing - This part integrates a desirable moral value in the lesson. Posttest - This measures how much you have learned from the entire module. EXPECTATIONS 1. Solve for the sides and diagonals of parallelograms. 2. Use properties to find measures of sides and diagonals involving parallelograms. 3. State the different properties of a parallelogram. PRETEST Directions: Read each of the following carefully. Choose the letter that corresponds to the correct answer. E A 1. What is the measure of side EA in parallelogram EASY A. 15 B. 20 C. 25 D. 30 Y S 15 2. If LOVE is a parallelogram and LR = 6 and L O R ER = 10, what is RV? E A. 6 B. 8 C. 10 V D. 12 M O For item numbers 3 – 5, use the parallelogram MORE 3. In a parallelogram MORE, ME = 2x + 3 and OR = 5x -12 what is the value of x? A. 3 B. 5 C. 12 E D. 15 4. In parallelogram MORE, what is the length of ME? A. 9 B. 10 C. 13 D. 25 5. In parallelogram MORE, if diagonal MR = 16, how long is MS? A. 4 B. 8 C. 12 S D. 16 R RECAP Let’s identify it!!! Direction: Write TRUE if the statement is correct; otherwise, write FALSE. __________1. In a parallelogram, only one pair of opposite sides are congruent. __________2. In a parallelogram, any two consecutive angles are complementary. __________3. In a parallelogram, any two opposite angles are congruent. __________4. A diagonal of a parallelogram divides the parallelogram in two congruent triangles. __________5. The diagonals of a parallelogram bisect each other. LESSON In the previous self-learning module, angles involving parallelogram are solved by applying the properties of parallelogram. Now we are going to find the measures of the lengths of unknown sides and diagonals in a parallelogram. Given: S □CARE is a parallelogram. ̅̅̅̅̅ 1. ̅̅̅̅ 𝐶𝐴 ≅ 𝐸𝑅 ̅̅̅̅ ≅ ̅̅̅̅̅ 𝐶𝐸 𝐸𝐴 Using the property of parallelogram in which any two opposite sides are congruent. ̅̅ ≅ ̅̅̅̅ 2. In diagonal CR, ̅̅ 𝐶𝑆 𝑆𝑅 In diagonal AE, ̅̅̅̅ 𝐸𝑆 ≅ ̅̅̅̅ 𝑆𝐴 Using the property of parallelogram in which the diagonals bisect each other. Examples: 1. □CALM is a parallelogram. If the length of ̅̅̅̅ = 4x-9 cm, 𝑀𝐿 ̅̅̅̅ = x+6 cm, 𝐶𝑀 ̅̅̅̅̅ = 3y-5 cm 𝐶𝐴 ̅̅̅̅ and 𝐴𝐿 = y+9 cm, find the value of each of the following: a. x b. y M ̅̅̅̅ c. 𝐶𝐴 d. ̅̅̅̅ 𝐴𝐿 e. Perimeter of □CALM C A S L Solutions: a. In □CALM, ̅̅̅̅ 𝐶𝐴 ≅ ̅̅̅̅̅ 𝑀𝐿 In parallelogram, opposite sides are congruent. 4x-9 = x+6 By substitution. 4x – x = 6 + 9 By Addition Property of Equality. 3x = 15 Divide each side of the equation by 3. x=5 ̅̅̅̅̅ ≅ ̅̅̅̅ b. In □CALM, 𝐶𝑀 𝐴𝐿 3y-5 = y+9 3y – y = 9+5 2y = 14 y=7 In parallelogram, opposite sides are congruent. By substitution. By Addition Property of Equality. Divide each side of the equation by 2. c. Since x = 5, then ̅̅̅̅ 𝐶𝐴 = ̅̅̅̅ = 𝐶𝐴 ̅̅̅̅ = 𝐶𝐴 ̅̅̅̅ 𝐶𝐴 = 4x – 9 4(5) – 9 20 – 9 11 cm d. Since y = 7, then ̅̅̅̅ 𝐴𝐿 = y + 9 ̅̅̅̅ = 7 + 9 𝐴𝐿 ̅̅̅̅ 𝐴𝐿= 16 cm ̅̅̅̅ = 11 cm, 𝑀𝐿 ̅̅̅̅ = 11 cm, 𝐴𝐿 ̅̅̅̅ = 16 cm, and 𝐶𝑀 ̅̅̅̅̅ = 16 cm, e. Since 𝐶𝐴 Therefore, the perimeter of □CALM is 54 cm. 2. Use the figure at the right to find the measures of the unknown variables and the indicated length of diagonals. Given: □CARE is a parallelogram, ̅̅̅̅ 𝐷𝐴 = 15, ̅̅̅̅ 𝐷𝐸 = 3y, ̅̅̅̅ = x – 3 and 𝐷𝑅 ̅̅̅̅ = 3x – 35 find: 𝐶𝐷 a. b. c. d. e. x y ̅̅̅̅ 𝐶𝐷 ̅̅̅̅ 𝐶𝑅 ̅̅̅̅ 𝐸𝐴 Solutions: a. In diagonal CR, ̅̅̅̅ 𝐶𝐷 ≅ ̅̅̅̅̅ 𝐷𝑅 In parallelogram, diagonals bisect each other. x – 3 = 3x – 35 By substitution. 3x – x = - 3 + 35 By Addition Property of Equality. 2x = 32 Divide each side of the equation by 2. x = 16 ̅̅̅̅ ≅ 𝐷𝐸 ̅̅̅̅̅ b. In diagonal AE, 𝐷𝐴 3y = 15 3y = 15 y=5 c. Since x = 16, then In parallelogram, diagonals bisect each other. By substitution. Divide each side of the equation by 3. ̅̅̅̅ = x – 3 𝐶𝐷 ̅̅̅̅ = 16 – 3 𝐶𝐷 ̅̅̅̅ 𝐶𝐷 = 13 ̅̅̅̅ = 13 and 𝐶𝐷 ̅̅̅̅ ≅ 𝐷𝑅 ̅̅̅̅̅ then ̅̅̅̅ ̅̅̅̅ + 𝐷𝑅 ̅̅̅̅̅ d. Since 𝐶𝐷 𝐶𝑅 = 𝐶𝐷 ̅̅̅̅ 𝐶𝑅 = 13 + 13 ̅̅̅̅ = 26 𝐶𝑅 ̅̅̅̅ = 15 and 𝐷𝐴 ̅̅̅̅ ≅ 𝐷𝐸 ̅̅̅̅̅ then ̅̅̅̅ ̅̅̅̅ + 𝐷𝐸 ̅̅̅̅̅ e. Since 𝐷𝐴 𝐸𝐴 = 𝐷𝐴 ̅̅̅̅ 𝐸𝐴 = 15 + 15 ̅̅̅̅ 𝐸𝐴 = 30 ACTIVITIES ACTIVITY 1: LET’S PRACTICE! Direction: Use the figure at the right to find the unknown variables, indicated sides and diagonals. ̅ = 𝑦 + 4 𝑐𝑚, ̅̅̅̅ □WISH is a parallelogram given: ̅̅̅̅ 𝑊𝐼 = 3𝑥 − 2 𝑐𝑚, 𝐼𝑆 𝑆𝐻 = 19 𝑐𝑚, ̅̅̅̅̅ = 10 cm, 𝐼𝑇 ̅̅̅ = 12 cm and 𝑆𝑇 ̅̅̅̅ = 9 cm . Find the value of each of the following: 𝑊𝐻 1. x 2. y 3. ̅̅̅̅ 𝐻𝑇 4. ̅̅̅̅̅ 𝑊𝑆 5. Perimeter of □WISH ACTIVITY 2: KEEP PRACTICING! Direction: In parallelogram HOPE, find the value of each of the unknown variables and the indicated sides. 1. x 2. y 3. ̅̅̅̅ 𝐸𝑃 4. ̅̅̅̅ 𝐸𝑂 5. ̅̅̅̅ 𝐻𝑃 ACTIVITY 3: TEST YOURSELF! Directions: Find the measure of the unknown variables and sides of the given parallelogram as shown in the figure below. Show your solution. 1. What is the value of x? 2. What is the value of y? 3. Find the perimeter of parallelogram FATE 4. If diagonal EA is 32, what is the measure of SE? 5. If FS is 11, how long is the length of diagonal FT? WRAP–UP Remember that… • To solve for the unknown sides and diagonals involving parallelograms, we need to be guided by the different properties of a parallelogram. • These are different Properties of Parallelogram that will guide you in solving for the sides and diagonals. 1. In a parallelogram, any two opposite sides are congruent. 2. The diagonals of a parallelogram bisect each other. VALUING REFLECTION: (Journal Writing) The properties of parallelogram involving sides and diagonals can be associated as the love of God given to us. We are all congruent and equal in the eyes of God. Just like diagonals, when we cross the road of life, we bisect each other and become equal on our own perspective. But sometimes we just need to be patient and wait for the perfect time to come for us to be successful in life. As a student, list down the different aspects of your life wherein you encounter that life is just like parallelogram considering the properties involving sides and diagonals. POSTTEST Directions: Read each of the following carefully. Choose the letter that corresponds to the correct answer. F A 25 1. What is the measure of side AS in parallelogram FAST A. 15 B. 20 C. 25 T 2. If LOVE is a parallelogram and LV = 14, L O R what is the length of LR? E A. 7 B. 8 S D. 30 C. 26 V D. 28 M O For item number 3 – 5, use the parallelogram MORE 3. In a parallelogram MORE, MO = 12x + 1 and ER = 2x +21 what is the value of x? A. 2 B. 5 C. 12 D. 15 4. In parallelogram MORE, what is the length of ER? A. 9 B. 10 C. 13 D. 25 5. In parallelogram MORE, if diagonal MS = 8, how long is MR? A. 4 B. 8 C. 12 D. 16 S E R 4. 18 cm ACTIVITY 1: LET’S PRACTICE! 1. 7 2. 6 3. 12 cm 4. TRUE 2. FALSE RECAP 1. FALSE 2. A PRETEST 1. A 3. B 3. TRUE 4. C 5. B 5. TRUE 5. 58 cm ACTIVITY 2: KEEP PRACTICING! 1. 3 2. 11 3. 20 4. 28 5. 24 ACTIVITY 3: TEST YOURSELF! 1. 4 2. 2 3. 76 4. 16 5. 22 POST TEST 1. C 2. A 3. A 4. D 5. D KEY TO CORRECTION References Alferez, Merle, and Alvin Lambino. Geometry. Quezon City: MSA Academic Institute, 2004. Bryant, Merden, Bulalayao, Leonides, Callanta, Melvin, Cruz, Jerry, De Vera, Richard, Garcia, Gilda, Javier, Sonia, Lazaro, Roselle, Mesterio, Bernadeth and Rommel Hero Saladino. Mathematics Grade 9 Learner’s Material. Sunshine Interlinks Publishing House, Inc., 2014. Oronce, Orlando, and Marilyn Mendoza. E-Math 9. Manila: Rex Book Store, Inc., 2015. Math is Fun post about Quadrilaterals. https://www.mathsisfun.com/ quadrilaterals.html (accessed July 14, 2020).
