Math Grade 8 Quarter 3 Solving Corresponding Parts of Congruent Triangles PDF

8 NAME:__________________________________________ GRADE/SECTION:______________________________ MATHEMATICS Quarter III – Week 5 Solving Corresponding Parts of Congruent Triangles CONTEXTUALIZED LEARNING ACTIVITY SHEETS SCHOOLS DIVISION OF PUERTO PRINCESA CITY...

8 NAME:__________________________________________ GRADE/SECTION:______________________________ MATHEMATICS Quarter III – Week 5 Solving Corresponding Parts of Congruent Triangles CONTEXTUALIZED LEARNING ACTIVITY SHEETS SCHOOLS DIVISION OF PUERTO PRINCESA CITY Mathematics – Grade 8 Contextualized Learning Activity Sheets (CLAS) Quarter III - Week 5: Solving Corresponding Parts of Congruent Triangles 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 the exploitation of such work for a 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 CLAS 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 Schools Division of Puerto Princesa City Development Team of the Contextualized Learning Activity Sheets Writer: Selah B. Cornelio Content Editor: Evelyn C. Silvestre Language Editor: Angelyn M. Aborot Proofreader: Irene M. Villamor Reviewers: Marie Vic C. Velasco PhD, Haydee C. Hitosis, Evelyn C. Silvestre, Joseph Aurello, and Maricar T. Caliwag Illustrator: Shiela Mae O. Salibio Layout Artist: Laarni F. Aballa Management Team: Servillano A. Arzaga, CESO V, SDS Loida P. Adornado, PhD, ASDS Cyril C. Serador, PhD, CID Chief Ronald S. Brillantes, EPS-LRMS Manager Marie Vic C. Velasco, PhD, EPS-Mathematics Eva Joyce C. Presto, PDO II Rhea Ann A. Navilla, Librarian II External Evaluator: Division Quality Assurance Team: Ronald S. Brillantes, Warly I. Salido, Haydee C. Hitosis, Joseph D. Aurello, Maricar T. Caliwag, and Meguilito D. Campillos Division of Puerto Princesa City-Learning Resource Management Section (LRMS) Sta. Monica Heights, Brgy. Sta. Monica, Puerto Princesa City Telephone No.: (048) 434 9438 Email Address: [email protected]  Lesson 1 Solving Corresponding Parts of Congruent Triangles MELC: Solves corresponding parts of congruent triangles. (M8GE-IIIf-1) Objectives: 1. Recall corresponding parts of congruent triangles. 2. Identify corresponding parts of congruent triangles. 3. Solve corresponding parts of congruent triangles. Let’s Try Directions: Read and analyze each item. Write the letter of the correct answer on the space provided before the number. __________1. How do you describe the corresponding parts of congruent triangles? A. opposite C. equilateral B. congruent D. acute __________2. Which of the following is NOT the corresponding parts of congruent triangles? A. Corresponding sides C. Corresponding angles B. Corresponding points D. Corresponding vertices For numbers 3 to 7, refer to the figure below. Given: ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍 __________3. Which statement is NOT true about corresponding sides? A. 𝐴𝐵 ≅ 𝑋𝑌 C. 𝐵𝐶 ≅ 𝑌𝑍 C. 𝐴𝐶 ≅ 𝑋𝑍 D. 𝐴𝐵 ≅ 𝑋𝑍 __________4. Which statement is NOT true about corresponding angles? A. ∠𝐴 ≅ ∠𝑌 C. ∠𝐴 ≅ ∠𝑋 B. ∠𝐵 ≅ ∠𝑌 D. ∠𝐶 ≅ ∠𝑍 __________5. What is the corresponding side of YZ ? A. AB C. BC B. AC D. ∠L __________6. If 𝐴𝐶 = 5𝑚, what is the measure of 𝑋𝑍 ? A. 3m C. 5m B. 10m D. 15m __________7. If ∠𝐶 = 60°, what is the measure of ∠𝑍 ? A. 40° C. 100° B. 60° D. 50° 1 __________8. These two triangles are congruent. If ∠C is 40°, what is the measure of ∠E? A. 90° C. 50° B. 40° D. 45° __________9. What is the value of x so that ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍? A. 1 C. 3 B. 2 D. 4 _________10. Given that ∆𝑃𝑄𝑅 ≅ ∆𝐿𝐽𝐾, PR = 2x + 6, JL = 10 units, LK = 3x – 6, what is the measure of side PR? A. 30 units C. 10 units B. 2 units D. 12 units Let’s Explore and Discover Let’s begin this lesson by reviewing congruent Unlocking of Difficulties triangles. As you go over the activities, keep this ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 - is read as question in mind, “When are two triangles congruent?” “triangle ABC is congruent to triangle DEF” What is meant by congruent triangles? ≅ - symbol for congruency How is this congruence determined? ∆ - symbol for triangle ↔ - symbol for Two line segments are correspondence, it means “to be congruent if their endpoints equivalent or similar” can be made to coincide. Two angles are congruent if their ∠𝐴 ≅ ∠𝐵 - means that the sides can be made to measure of angle A is equal to coincide. Similarly, two the measure of angle B. triangles are congruent if all For example: ∠𝐴 = 30° their parts can be made to ∠𝐵 = 30° coincide. These imply that 𝐴𝐵 ≅ 𝐶𝐷 - means that the two triangles to be measure of segment AB is equal congruent, they must have to the measure of segment CD. the same shape and the For example: 𝐴𝐵 = 20𝑐𝑚 same size. 𝐶𝐷 = 20𝑐𝑚 2 Remember Two triangles are congruent if and only if their vertices can be paired so that corresponding sides are congruent and corresponding angles are congruent. (Source: Julieta G. Bernabe, Soledad Jose-Dilao, Ed.D., and Fernando B. Orines, Geometry Textbook for Third Year, Pasig City: DepEd-IMCS, 2009, 89.) 1.1 Corresponding Parts of Congruent Triangles (A Recall) Example 1: Given ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 Suppose ∆𝐴𝐵𝐶 is made to coincide with ∆𝐷𝐸𝐹 such that the vertices of ∆𝐴𝐵𝐶 fit exactly over the vertices of ∆𝐷𝐸𝐹. The corresponding vertices of two triangles are: 𝐴 ↔𝐷 𝐵 ↔𝐸 𝐶 ↔𝐹 The congruent corresponding angles and congruent corresponding sides are marked identically. Then, we can say that: Corresponding Angles Corresponding Sides ∠𝐴 ≅ ∠𝐷 𝐴𝐵 ≅ 𝐷𝐸 ∠𝐵 ≅ ∠𝐸 𝐵𝐶 ≅ 𝐸𝐹 ∠𝐶 ≅ ∠𝐹 𝐶𝐴 ≅ 𝐹𝐷 Example 2: Given ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷 These are the six congruent corresponding parts of congruent triangles. The corresponding vertices are: 𝐴 ↔𝐶 𝐵 ↔𝐵 𝐷 ↔𝐷 Corresponding Corresponding Angles Sides ∠𝐴 ≅ ∠𝐶 𝐴𝐵 ≅ 𝐶𝐵 ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐵𝐷 𝐴𝐷 ≅ 𝐶𝐷 ∠𝐴𝐷𝐵 ≅ ∠𝐶𝐷𝐵 𝐵𝐷 ≅ 𝐵𝐷 (Source: Emmanuel P. Abuzo et al., Mathematics Learner’s Module 8, Pasig City: DepEd-IMCS, 2013, 352-353.) 3 1.2 Solving Corresponding Parts of Congruent Triangles Remember 1.2 Solving Corresponding Corresponding Parts parts of of Congruent congruent trianglesTriangles are congruent. (CPCTC) Example 1: Given ∆𝐴𝐵𝐶 ≅ ∆ 𝐷𝐸𝐹, find the missing measures of each corresponding parts of these congruent triangles. Step 1: Write the corresponding vertices: ∆𝐴𝐵𝐶 ≅ ∆ 𝐷𝐸𝐹. 𝐴 ↔𝐷 𝐵 ↔𝐸 𝐶 ↔𝐹 Step 2: Name the congruent corresponding angles and congruent corresponding sides. Corresponding Angles Corresponding Sides ∠𝐴 ≅ ∠𝐷 𝐴𝐵 ≅ 𝐷𝐸 ∠𝐵 ≅ ∠𝐸 𝐵𝐶 ≅ 𝐸𝐹 ∠𝐶 ≅ ∠𝐹 𝐶𝐴 ≅ 𝐹𝐷 Step 3: Identify the missing parts. To answer this, just remember the congruent corresponding angles and congruent corresponding sides. Corresponding Angles Corresponding Sides 𝑚∠𝐴 = _______ 𝐷𝐸 = ________ 𝑚∠𝐵 = _______ 𝐸𝐹 = ________ 𝑚∠𝐶 = _______ 𝐶𝐴 𝑜𝑟 𝐴𝐶 = ________ Step 4: Apply the CPCTC. 𝑚∠𝐴 = ____73°___ 𝐷𝐸 = ____2.6𝑐𝑚____ 𝑚∠𝐵 = ____65°___ 𝐸𝐹 = _____3.7𝑐𝑚___ 𝑚∠𝐶 = ____42°___ 𝐶𝐴 𝑜𝑟 𝐴𝐶 = ____3.5𝑐𝑚____ Example 2: Given ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐵𝐶. Find the value of x. Step 1: Write the corresponding vertices: ∆𝐴𝐵𝐶 ≅ ∆ 𝐷𝐵𝐶 𝐴 ↔𝐷 𝐵 ↔𝐵 𝐶 ↔𝐶 4 Step 2: Name the congruent corresponding angles and congruent corresponding sides. Corresponding Angles Corresponding Sides ∠𝐴 ≅ ∠𝐷 𝐴𝐵 ≅ 𝐷𝐵 ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐵𝐶 𝐴𝐶 ≅ 𝐷𝐶 ∠𝐴𝐶𝐵 ≅ ∠𝐷𝐶𝐵 𝐵𝐶 ≅ 𝐵𝐶 Step 3: Solve for the value of x. Since, 𝐴𝐵 ≅ 𝐷𝐵 Given 2𝑥 − 4 = 𝑥 + 5 Substitution Method 2𝑥 + (−𝑥) − 4 + (4) = 𝑥 + (−𝑥) + 5 + (4) Addition Property of Equality 𝑥=9 Solve To check, substitute the value of x to each of the given measures. 𝐴𝐵 = 2𝑥 − 4 𝐷𝐵 = 𝑥 + 5 𝐴𝐵 = 2(9) − 4 𝐷𝐵 = (9) + 5 𝐴𝐵 = 18 − 4 𝐷𝐵 = 14 𝐴𝐵 = 14 The value of x is 9. Therefore, 𝐴𝐵 ≅ 𝐷𝐵. Example 3: Given ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍, find the value of x and y. Step 1: Write the corresponding vertices: ∆𝐴𝐵𝐶 ≅ ∆ 𝑋𝑌𝑍. 𝐴 ↔𝑋 𝐵 ↔𝑌 𝐶 ↔𝑍 Step 2: Name the congruent corresponding angles and congruent corresponding sides. Corresponding Angles Corresponding Sides ∠𝐴 ≅ ∠𝑋 𝐴𝐵 ≅ 𝑋𝑌 ∠𝐵 ≅ ∠𝑌 𝐵𝐶 ≅ 𝑌𝑍 ∠𝐶 ≅ ∠𝑍 𝐴𝐶 ≅ 𝑋𝑍 Step 3: Solve for the value of x. Since, 𝐴𝐵 ≅ 𝑋𝑌 Given 3𝑥 + 8 = 5𝑥 Substitution Method 2𝑥 + (−5𝑥) + 8 + (−8) = 5𝑥 + (−5𝑥) + (−8) Addition Property of Equality −2𝑥 = −8 Solve = Division Property of Equality 𝑥= 4 Solve Step 3: Solve for the value of y. Since, ∠𝐴 ≅ ∠𝑋 Given 𝑚∠𝐴 = 𝑚∠𝑋 5𝑦 + 11 = 6𝑦 + 2 Substitution Method 5𝑦 + (−6𝑦) + 11 + (−11) = 6𝑦 + (−6𝑦) + 2 + (−11) Addition Property of Equality −𝑦 = −9 Solve = Division Property of Equality 𝑦= 9 Solve 5 To check, substitute 4, as the value of x, to each of the given measures. 𝐴𝐵 = 3𝑥 + 8 𝑋𝑌 = 5𝑥 𝐴𝐵 = 3(4) + 8 𝑋𝑌 = 5(4) 𝐴𝐵 = 12 + 8 𝑋𝑌 = 20 𝐴𝐵 = 20 The value of x is 4. Therefore, 𝐴𝐵 ≅ 𝑋𝑌. To check, substitute 9, as the value of y, to each of the given measures. 𝑚∠𝐴 = 5𝑦 + 11 𝑚∠𝑋 = 6𝑦 + 2 𝑚∠𝐴 = 5(9) + 11 𝑚∠𝑋 = 6(9) + 2 𝑚∠𝐴 = 45 + 11 𝑚∠𝑋 = 54 + 2 𝑚∠𝐴 = 56 𝑚∠𝑋 = 56 The value of y is 9. Therefore, ∠𝐴 ≅ ∠𝑋. (Source: Emmanuel P. Abuzo et al., Mathematics Learner’s Module 8, Pasig City: DepEd-IMCS, 2013, 352-353.) Let’s Practice Directions: Based on the given congruent triangles in the first column, provide the matches of the given corresponding sides and corresponding angles. Write your answers in the appropriate columns. ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 Corresponding angles Corresponding sides ∠𝐴 ≅ _______ ________ ≅ 𝐷𝐸 _______ ≅ _________ _______ ≅ ________ ________ ≅ _________ ________ ≅ _______ Directions: Solve for the measures of the corresponding parts of congruent triangles. Write your answers on the given blanks. Given ∆𝐾𝐹𝐶 ≅ ∆𝑀𝐴𝐷 𝑚∠𝐾 = _____________ 𝐾𝐹 = ____________ 16 cm 𝑚∠𝐴 = _____________ 𝐴𝐷 = ____________ 𝑚∠𝐷 = _____________ 𝑀𝐷 = ____________ 1. How many pairs of corresponding parts are congruent if two triangles are congruent? __________________________________________________________________________________ 2. When can you say that the two triangles are congruent? __________________________________________________________________________________ 6  Let’s Do More Directions: Solve for the value of x and determine the measures of each corresponding part of the congruent triangles. Write your answer on the given blanks. Show your solutions. Given: ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐵𝐷. 𝐴𝐶 = 5𝑥, 𝐴𝐷 = 𝑥 + 16, 𝐵𝐶 = 2𝑥 + 4, 𝐵𝐷 = 𝑥 + 8, 𝑎𝑛𝑑 𝐴𝐵 = 4𝑥 𝐴𝐶 ≅ 𝐴𝐷 = __________ 𝐵𝐶 ≅ 𝐵𝐷 = __________ 𝐴𝐵 ≅ 𝐴𝐵 = __________ Directions: Solve the following corresponding parts of congruent triangles to find the values of x and y. Show your solutions on the provided space. Given: ∆𝐿𝑀𝑁 ≅ ∆𝑆𝑅𝑄, find the values of x and y. 1. How can the corresponding parts of congruent triangles be applied or be used to solve real-life situations? ________________________________________________________________________________________ ________________________________________________________________________________________ 2. Why is it important to know the correspondence of things in our surroundings? ________________________________________________________________________________________ ________________________________________________________________________________________ 7  Let’s Sum It Up I. Directions: Complete the statements by writing the appropriate term/s from the box. Write these on the spaces provided. Corresponding sides Corresponding angles Congruent Congruent triangles Vertices Correspondence For this lesson, I have learned the following: 1. Two triangles are congruent if and only if their vertices can be paired so that _____________________________ are congruent and _____________________________are congruent. 2. Corresponding parts of congruent triangles are ________________________________. 3. In order to solve corresponding parts of__________________________________, you need to apply the CPCTC. 4. The __________________________ and _________________________ are the corresponding parts of congruent triangles. II. Directions: Encircle the 10 important words in this module about Solving Corresponding Parts of Congruent Triangles in the Word Search Puzzle below. You can find it HORIZONTALLY, VERTICALLY and DIAGONALLY. 8  Let’s Assess Directions: Read and analyze each item. Write the letter of the correct answer on the space provided before the number. __________1. Which of the following is NOT the corresponding parts of congruent triangles? A. Corresponding sides C. Corresponding angles B. Corresponding points D. Corresponding vertices __________2. How do you describe the corresponding parts of congruent triangles? A. opposite C. equilateral B. congruent D. acute For numbers 3 to 7, refer to the figure at the right. Given: ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍 __________3. Which statement is NOT true about corresponding sides? A. 𝐴𝐵 ≅ 𝑋𝑌 C. 𝐵𝐶 ≅ 𝑌𝑍 C. 𝐴𝐶 ≅ 𝑋𝑍 D. 𝐴𝐵 ≅ 𝑋𝑍 __________4. Which statement is NOT true about corresponding angles? A. ∠𝐴 ≅ ∠𝑋 C. ∠𝐴 ≅ ∠𝑌 B. ∠𝐵 ≅ ∠𝑌 D. ∠𝐶 ≅ ∠𝑍 __________5. What is the corresponding side of YZ ? A. 𝐵𝐶 B. 𝐴𝐶 C. 𝐴𝐵 D. ∠L __________6. If 𝐴𝐶 = 5𝑚, what is the measure of 𝑋𝑍 ? A. 3m C. 10m B. 5m D. 15m __________7. If ∠𝐶 = 60°, what is the measure of ∠𝑍 ? A. 40° B. 70° C. 60° D. 50° __________8. These two triangles at the right are congruent. If ∠C is 40°, what is the measure of ∠E? A. 90° C. 50° B. 40° D. 45° __________9. What is the value of x so that ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍? A. 1 C. 3 B. 2 D. 4 __________10. Given that ∆𝑃𝑄𝑅 ≅ ∆𝐿𝐽𝐾, PR = 2x + 6, JL = 10 units, LK = 3x – 6, what is the measure of side PR? A. 30 units C. 10 units B. 2 units D. 12 units 9  Answer Key Let’s Try Let’s Do More 1. B 2. B 3. D 4. A 5. C Since x=4, substitute it 6. C 7. B 8. B 9. A 10. A to all of the measures: To solve for x: 1. AC=5x 2. BD=x + 8 𝐴𝐶 ≅ 𝐴𝐷 Let’s Practice =5(4) =(4)+8 5𝑥 = 𝑥 + 16 =20 =12 5𝑥 + (−𝑥) = 𝑥 + (−𝑥) + 16 4𝑥 = 16 3. AB=4x=4(4) = 16 Corresponding Corresponding 𝑥=4 Therefore, Angles Sides 𝐴𝐶 ≅ 𝐴𝐷 = 20 𝐵𝐶 ≅ 𝐵𝐷 = 12 ∠𝐴 ≅ ∠𝐷 𝐴𝐵 ≅ 𝐷𝐸 𝐴𝐵 ≅ 𝐴𝐵 = 16 ∠𝐵 ≅ ∠𝐸 𝐴𝐶 ≅ 𝐷𝐹 ∠𝐶 ≅ ∠𝐹 𝐵𝐶 ≅ 𝐸𝐹 To solve for x: Since 𝐿𝑁 ≅ 𝑆𝑄 1. 𝑚 < 𝐾 = 30° 𝐾𝐹 = 16𝑐𝑚 2𝑥 + 3 = 31 2. 𝑚 < 𝐴 = 90° 𝐴𝐷 = 10𝑐𝑚 2𝑥 + 3 + (−3) = 31 + (−3) To solve for y: 3. 𝑚 < 𝐷 = 60° 𝑀𝐷 = 25cm 2𝑥 = 28 Since 2𝑥 28 < 𝑁 ≅< 𝑄 = 𝑚