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Here are the key points in the example: 1. The triangles ΔCAT and ΔDOG are placed on top of each other so that their vertices coincide or overlap. 2. This establishes a correspondence between the parts of the two triangles. 3. Since the vertices coincide, the corresponding sides and angles also coincide. 4. Triangles with coinciding corresponding parts are said to be congruent. 5. The congruence of the two triangles is symbolized as ΔCAT ≅ ΔDOG.

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CO_Q3_Mathematics 8_ Module 3 Mathematics Quarter 3 – Module 3 Illustrating Triangle Congruence
Mathematics – Grade 8 Alternative Delivery Mode Quarter 3 – Module 3: Illustrating Triangle Congruence 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 Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio Printed in the Philippines __________________________ Department of Education – Caraga Region Office Address: Learning Resource Management Section (LRMS)tr555555 J.P. Rosales Avenue, Butuan City, Philippines 8600 Telefax Nos.: (085) 342-8207 / (085) 342-5969 E-mail Address: caraga@deped.gov.ph Development Team of the Module Writer: Rochelle Anna Myla A. Paronda, Language Editor: Joel B. Asonto Content Editors: Victorino Nimes, Myracell Buenaflor, Marie Grace O. Aparre, Ella C. Armayan Reviewers: Rhea J. Yparraguirre, Villaflor D. Edillor, Shirley C. Compe, Imelda B. Llasos, Caren L. Alamin, Jayrold Virtudazo Navarra, Alma R. Velaasco Illustrator: Fritch A. Paronda Layout Artists: Fritch A. Paronda, James B. De Los Reyes Management Team: Francis Cesar B. Bringas Isidro M. Biol, Jr. Maripaz F. Magno Josephine Chonie M. Obseñares Josita B. Carmen Celsa A. Casa Regina Euann A. Puerto Bryan L. Arreo Elnie Anthony P. Barcena Leopardo P. Cortes, Jr. Claire Ann P. Gonzaga
8 Mathematics Quarter 3 – Module 3 Illustrating Triangle Congruence
Introductory Message This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson. Each SLM is composed of different parts. Each part shall guide you step-by- step as you discover and understand the lesson prepared for you. Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self- check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these. In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning. Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task. If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Thank you.
1 CO_Q3_Mathematics 8_ Module 3 What I Need to Know This module was designed and written with you in mind. It is here to help you illustrate triangle congruence. You are provided with varied activities to process the knowledge and skills learned and to deepen and transfer your understanding of the lesson. The scope of this module enables you to use it in many different learning situations. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using. This module contains concepts on how to illustrate triangle congruence. After going through this module, you are expected to: 1. define triangle congruence; 2. draw and label the corresponding parts of two congruent triangles; 3. identify corresponding parts of two congruent triangles; and 3. relate triangle congruence in real-life.
2 CO_Q3_Mathematics 8_ Module 3 What I Know Directions: Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper. 1. Which of the following is a symbol used for congruence? A. = C. ≈ B. ≠ D. ≅ 2. How many pairs of corresponding congruent angles are there in two congruent triangles? A. 1 C. 3 B. 2 D. 4 3. Which of the symbols below is used to indicate correspondence? A. ≅ C. 𝛥 B. ∠ D. ↔ 4. Which of the following statements best describe congruent triangles? I. They are similar. II. They have the same size. III. They have the same shape. IV. They are equilateral triangles. A. I only C. III and IV together B. III only D. II and III together 5. How many pairs of corresponding sides are there in two congruent triangles? A. 1 C. 3 B. 2 D. 4 6. Which of the following statements best describe the corresponding parts of congruent triangles? A. They are not equal. B. They are congruent. C. They are supplementary. D. They are complementary.
3 CO_Q3_Mathematics 8_ Module 3 For items 7 - 10, consider the figure below. 7. Which segment corresponds to 𝑁𝑇 ̅̅̅̅ ? A. 𝐴𝑇 ̅̅̅̅ C. 𝐶𝑀 ̅̅̅̅̅ B. 𝐴𝐶 ̅̅̅̅ D. 𝑀𝐴 ̅̅̅̅̅ 8. Which angle corresponds to ∠𝑇 ? A. ∠𝐴 C. ∠𝑀 B. ∠𝐶 D. ∠𝑁 9. Which vertex corresponds to 𝐴? A. 𝐴 C. 𝑁 B. 𝑀 D. 𝑇 10.What conclusion can you make in the figure? A. 𝛥𝐴𝑁𝑇 ≅ 𝛥𝑀𝐴𝐶 B. 𝛥𝐴𝑁𝑇 ≅ 𝛥𝐶𝐴𝑀 C. 𝛥𝐴𝑀𝐶 ≅ 𝛥𝐴𝑇𝑁 D. 𝛥𝐴𝑀𝐶 ≅ 𝛥𝑇𝐴𝑁 11.If 𝛥𝐿𝑀𝑁 ≅ 𝛥𝑃𝑅𝑄, then what angle corresponds to ∠𝑄 ? A. ∠𝐿 C. ∠𝑃 B. ∠𝑁 D. ∠𝑅 12.Which of the following figures illustrate pair of congruent triangles? A. C. B. D. 𝑀 𝑁 𝐶 𝑇 𝐴 A D C B E O P R S T Q U V W A B C D
4 CO_Q3_Mathematics 8_ Module 3 13.Which of the following correspondence is equivalent to ∠𝐽𝐾𝐿 ≅ ∠𝑀𝑁𝑂? A. ∠𝐾𝐽𝐿 ↔ ∠𝑀𝑂𝑁 B. ∠𝐾𝐽𝐿 ↔ ∠𝑂𝑀𝑁 C. ∠𝐽𝐿𝐾 ↔ ∠𝑁𝑀𝑂 D. ∠𝐿𝐾𝐽 ↔ ∠𝑂𝑁𝑀 14. If 𝛥𝑁𝐴𝑇 ≅ 𝛥𝐺𝐸𝑂, then name the corresponding sides of the congruent triangles. A. 𝑁𝐴 ̅̅̅̅ ≅ 𝐺𝐸 ̅̅̅̅ C. 𝐴𝑇 ̅̅̅̅ ≅ 𝐸𝑂 ̅̅̅̅ 𝐴𝑇 ̅̅̅̅ ≅ 𝐸𝑂 ̅̅̅̅ 𝑁𝑇 ̅̅̅̅ ≅ 𝐸𝐺 ̅̅̅̅ 𝑁𝑇 ̅̅̅̅ ≅ 𝐺𝑂 ̅̅̅̅ 𝑁𝐴 ̅̅̅̅ ≅ 𝑂𝐺 ̅̅̅̅ B. 𝑁𝐴 ̅̅̅̅ ≅ 𝐺𝑂 ̅̅̅̅ D. 𝐴𝑇 ̅̅̅̅ ≅ 𝐺𝐸 ̅̅̅̅ 𝐴𝑇 ̅̅̅̅ ≅ 𝐸𝑂 ̅̅̅̅ 𝑁𝐴 ̅̅̅̅ ≅ 𝐸𝑂 ̅̅̅̅ 𝑁𝑇 ̅̅̅̅ ≅ 𝐺𝐸 ̅̅̅̅ 𝑁𝑇 ̅̅̅̅ ≅ 𝐺𝑂 ̅̅̅̅ 15.Lyka was asked by her Math teacher to give and write the corresponding sides of congruent triangles, 𝛥𝐶𝐴𝑇 and 𝛥𝐷𝑂𝐺, on the board. Her answer is shown below. 𝐴𝐶 ̅̅̅̅ ↔ 𝑂𝐷 ̅̅̅̅ 𝐶𝑇 ̅̅̅̅ ↔ 𝐷𝐺 ̅̅̅̅ 𝐴𝑇 ̅̅̅̅ ↔ 𝑂𝐷 ̅̅̅̅ Is Lyka’s answer correct? A. Yes, because each side of 𝛥𝐶𝐴𝑇 is paired correctly to each side of 𝛥𝐷𝑂𝐺. B. Yes, because each angle of 𝛥𝐶𝐴𝑇 is paired correctly to each angle of 𝛥𝐷𝑂𝐺. C. No, because 𝐴𝑇 ̅̅̅̅ corresponds to 𝑂𝐺 ̅̅̅̅. D. No, because 𝐶𝑇 ̅̅̅̅ corresponds to 𝑂𝐺 ̅̅̅̅.
5 CO_Q3_Mathematics 8_ Module 3 Lesson 1 Illustrating Triangle Congruence Designs and patterns having the same size and shape thrive in almost every place. These can be seen in bridges and condominiums, furniture and appliances, as well as in fabrics and crafts. Triangle is one of the most important figures studied in Mathematics because of its many uses in the real-world. Let us know more about triangles, particularly congruent triangles by performing the array of activities in this module. Have fun! What’s In Activity: Identify V-A-S (Vertices – Angles – Sides) Directions: Identify the vertices, angles, and sides of each triangle. Then, answer the questions below using complete sentences. Questions: 1. How did you identify the vertices, angles, and sides of each triangle? 2. How would you define angle? 3. What about the definition of a vertex? 4. In geometry, what is meant by side? Vertices Angles Sides Vertices Angles Sides 𝐵 𝐴 𝐶 𝑅 𝑄 𝑆
6 CO_Q3_Mathematics 8_ Module 3 What’s New Activity: Four Pics – Two Words Directions: Consider the figures below and answer the questions that follow. Questions 1. What geometric figures were highlighted in the illustrations? 2. What have you observed with the size and shape of the geometric figures? Explain your answer. What is It  Two line segments are congruent if their endpoints can be made to coincide.  Two angles are congruent if their sides can be made to coincide.  Two triangles are equal if they have the same size and shape.  Two triangles are congruent if their corresponding parts are congruent.
7 CO_Q3_Mathematics 8_ Module 3 Assume 𝛥𝑁𝐴𝑇 coincides with 𝛥𝐺𝐸𝑂, such that the vertices of 𝛥𝑁𝐴𝑇 fit exactly over the vertices of 𝛥𝐺𝐸𝑂 . The correspondence can be created between triangles. 𝛥𝑁𝐴𝑇 ↔ 𝛥𝐺𝐸𝑂 In this correspondence, there are three pairs of corresponding vertices, angles, and sides. Since the vertices of 𝛥𝑁𝐴𝑇 fit exactly with the vertices of 𝛥𝐺𝐸𝑂, therefore 𝛥𝑁𝐴𝑇 and 𝛥𝐺𝐸𝑂 are congruent. In symbols, 𝛥𝑁𝐴𝑇 ≅ 𝛥𝐺𝐸𝑂 Definition of Correspondence Pairing the parts of one group to the parts of another group is called correspondence. Correspondence uses the notation “↔”. Corresponding Angles ∠𝑁 ↔ ∠𝐺 ∠𝐴 ↔ ∠𝐸 ∠𝑇 ↔ ∠𝑂 𝐴 𝑁 𝑇 𝐸 𝐺 𝑂 Corresponding Sides 𝑁𝐴 ̅̅̅̅ ↔ 𝐺𝐸 ̅̅̅̅ 𝐴𝑇 ̅̅̅̅ ↔ 𝐸𝑂 ̅̅̅̅ 𝑁𝑇 ̅̅̅̅ ↔ 𝐺𝑂 ̅̅̅̅ Corresponding Vertices 𝑁 ↔ 𝐺 𝐴 ↔ 𝐸 𝑇 ↔ 𝑂 Note In naming congruent triangles, the order of the letters representing the vertices of one triangle shall be written in the same order as their corresponding vertices in the other triangle. Definition of Congruent Triangles Two triangles are congruent if and only if the corresponding sides and the corresponding angles are congruent.
8 CO_Q3_Mathematics 8_ Module 3 (𝛥𝑁𝐴𝑇 ≅ 𝛥𝐺𝐸𝑂 is read as “triangle 𝑁 − 𝐴 − 𝑇 is congruent to triangle 𝐺 − 𝐸 − 𝑂”) It implies that the corresponding parts of congruent triangles are also congruent. Conversely, if the corresponding parts of the two triangles are congruent, then the two triangles are congruent. Example 1 𝛥𝐶𝐴𝑇 ≅ 𝛥𝐷𝑂𝐺. Write down the pairs of corresponding parts. Solution The congruent parts of congruent triangles can be determined with the help of arcs and tick-marks. CPCTC Corresponding Parts of Congruent Triangles are Congruent. Congruent Angles ∠𝑁 ≅ ∠𝐺 ∠𝐴 ≅ ∠𝐸 ∠𝑇 ≅ ∠𝑂 Congruent Sides 𝑁𝐴 ̅̅̅̅ ≅ 𝐺𝐸 ̅̅̅̅ 𝐴𝑇 ̅̅̅̅ ≅ 𝐸𝑂 ̅̅̅̅ 𝑁𝑇 ̅̅̅̅ ≅ 𝐺𝑂 ̅̅̅̅ Corresponding Sides 𝐶𝐴 ̅̅̅̅ ↔ 𝐷𝑂 ̅̅̅̅ 𝐶𝑇 ̅̅̅̅ ↔ 𝐷𝐺 ̅̅̅̅ 𝐴𝑇 ̅̅̅̅ ↔ 𝑂𝐺 ̅̅̅̅ Corresponding Angles ∠𝐶 ↔ ∠𝐷 ∠𝐴 ↔ ∠𝑂 ∠𝑇 ↔ ∠𝐺 𝐴 𝐶 𝑇 𝑂 𝐷 𝐺
9 CO_Q3_Mathematics 8_ Module 3 Example 2 Given that the two triangles below are congruent, list the pairs of congruent angles and congruent sides and then name the congruent triangles. Solution 𝛥𝑃𝑆𝐴 ≅ 𝛥𝑁𝑆𝑂 Example 3 Consider 𝛥𝐹𝐴𝐸 ≅ 𝛥𝑇𝐸𝐴. List down the pairs of congruent angles and congruent sides. Solution Again, the congruent parts of congruent triangles can be determined with the help of arcs and tick-marks. Since 𝛥𝐹𝐴𝐸 ≅ 𝛥𝑇𝐸𝐴, then the corresponding parts are as follows: 𝐸 𝑇 𝐹 𝐴 Corresponding Angles ∠𝐹 ↔ ∠𝑇 ∠𝐹𝐸𝐴 ↔ ∠𝑇𝐴𝐸 ∠𝐹𝐴𝐸 ↔ ∠𝑇𝐸𝐴 Corresponding Sides 𝐹𝐴 ̅̅̅̅ ↔ 𝑇𝐸 ̅̅̅̅ 𝐹𝐸 ̅̅̅̅ ↔ 𝑇𝐴 ̅̅̅̅ 𝐸𝐴 ̅̅̅̅ ↔ 𝐴𝐸 ̅̅̅̅ 𝑃 𝐴 𝑆 𝑁 𝑂 Congruent Angles ∠𝐴 ≅ ∠𝑂 ∠𝑃 ≅ ∠𝑁 ∠𝑃𝑆𝐴 ≅ ∠𝑁𝑆𝑂 Congruent Sides 𝑃𝑆 ̅̅̅̅ ≅ 𝑁𝑆 ̅̅̅̅ 𝑃𝐴 ̅̅̅̅ ≅ 𝑁𝑂 ̅̅̅̅ 𝑆𝐴 ̅̅̅̅ ≅ 𝑆𝑂 ̅̅̅̅
10 CO_Q3_Mathematics 8_ Module 3 Example 4 Assume that segments and angles in the numbered triangles are congruent. Identify which triangles are congruent. Solution Looking at the figures, it can be deduced that 𝛥1 ≅ 𝛥6, 𝛥2 ≅ 𝛥5, and 𝛥3 ≅ 𝛥4. Remember:  Triangles that have exactly the same size and shape are congruent.  Two triangles are congruent if their corresponding sides and angles are congruent.  Two triangles are congruent if they fit exactly or coincide when one triangle is put on top of the other. 1 2 3 4 5 6
11 CO_Q3_Mathematics 8_ Module 3 What’s More Activity 1: Find My PARTner Description: In this activity, you will be pairing parts of one triangle to the parts of another triangle. Note that the notation “↔” shows corresponding parts. Directions: Copy the table on your answer sheet and supply the missing part of the triangle that will make each correspondence complete. Answer the questions that follow. Corresponding Vertices Corresponding Angles Corresponding Sides 𝐴 ↔ _____ _____ ↔ 𝐸 𝐶 ↔ 𝐹 ∠𝐴 ↔ _____ ∠𝐵 ↔ _____ ∠𝐶 ↔ _____ 𝐴𝐵 ̅̅̅̅ ↔ _____ _____ ↔ 𝐷𝐹 ̅̅̅̅ 𝐵𝐶 ̅̅̅̅ ↔ _____ Questions: 1. What have you noticed on the given triangles? 2. What does the mark of the angles and sides of the triangle indicate? 3. Are there angles and sides that have the same measure and length? Explain your answer. 4. Describe the relationship between 𝛥𝐴𝐵𝐶 and𝛥𝐷𝐸𝐹. 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹
12 CO_Q3_Mathematics 8_ Module 3 Activity 2: Draw and Name Directions: Do what is asked. Use a separate sheet of paper. 1. Draw 𝛥𝐴𝐶𝑇 ≅ 𝛥𝑁𝑂𝑊. Put identical markings on corresponding congruent parts. 2. 𝛥𝑆𝑇𝑅 ≅ 𝛥𝑅𝐴𝑆. Name the six pairs of corresponding congruent parts. 3. Listed below are the six pairs of corresponding parts of two congruent triangles. Name the two congruent triangles. Congruent Angles Congruent Sides ∠𝑇 ≅ ∠𝐸 ∠𝑂 ≅ ∠𝑂 ∠𝑊 ≅ ∠𝑁 𝑇𝑊 ̅̅̅̅̅ ≅ 𝐸𝑁 ̅̅̅̅ 𝑂𝑇 ̅̅̅̅ ≅ 𝑂𝐸 ̅̅̅̅ 𝑊𝑂 ̅̅̅̅̅ ≅ 𝑁𝑂 ̅̅̅̅ 𝑇 𝐴 𝑆 𝑅 Congruent Angles Congruent Sides Congruent Triangles
13 CO_Q3_Mathematics 8_ Module 3 What I Have Learned Let us see how far you have learned about congruent triangles by answering each question in the diagram below. Consider the figure at the center. Write your answers on a separate paper. What are the congruent triangles? What are the corresponding vertices? What are the corresponding angles? What are the corresponding sides? What are the congruent angles? What are the congruent sides? 𝐸 𝑅 𝐶 𝑂 𝑀
14 CO_Q3_Mathematics 8_ Module 3 What I Can Do One of the structural applications of triangles can be seen in bridges. In the illustration below, there are pairs of congruent triangles. Do what is asked below and write your answer on a separate paper. Directions: 1. Select a pair of congruent triangles. 2. Label these triangles as 𝛥𝑆𝐴𝐹 and 𝛥𝐸𝐴𝐹. 3. Put similar markings to show clearly the congruent sides and congruent angles. 4. List down the corresponding congruent parts of the congruent triangles.
15 CO_Q3_Mathematics 8_ Module 3 Assessment Directions: Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper. 1. Which of the following statements is not true about congruent triangles? A. They are dissimilar. B. They are coinciding. C. They have the same size and shape. D. They have corresponding congruent parts. 2. How many pairs of corresponding congruent parts are there in two congruent triangles? A. 2 C. 4 B. 3 D. 6 3. If 𝛥𝐹𝑅𝑌 ≅ 𝛥𝐻𝑂𝑇, which segment is congruent to 𝑅𝑌 ̅̅̅̅? A. 𝐻𝑇 ̅̅̅̅ C. 𝑂𝑇 ̅̅̅̅ B. 𝐹𝑌 ̅̅̅̅ D. 𝑅𝐹 ̅̅̅̅ 4. Which of the following is true about the corresponding parts of congruent triangles? A. They are unequal. B. They are congruent. C. They are supplementary. D. They are complementary. 5. If 𝛥𝑆𝐼𝑇 ≅ 𝛥𝐻𝑂𝑃, then what angle corresponds to ∠𝑇? A. ∠𝐻 C. ∠𝑃 B. ∠𝐼 D. ∠𝑆 6. Name the congruent triangles whose pairs of corresponding parts are shown below. ∠𝐻 ≅ ∠𝑆 𝑅𝐸 ̅̅̅̅ ≅ 𝑁𝑂 ̅̅̅̅ ∠𝑅 ≅ ∠𝑁 𝐻𝐸 ̅̅̅̅ ≅ 𝑆𝑂 ̅̅̅̅ ∠𝐸 ≅ ∠𝑂 𝐻𝑅 ̅̅̅̅ ≅ 𝑆𝑁 ̅̅̅̅ A. 𝛥𝑁𝑂𝑆 ≅ 𝛥𝐻𝐸𝑅 C. 𝛥𝑆𝑂𝑁 ≅ 𝛥𝐸𝐻𝑅 B. 𝛥𝑁𝑆𝑂 ≅ 𝛥𝐸𝑅𝐻 D. 𝛥𝑆𝑂𝑁 ≅ 𝛥𝐻𝐸𝑅
16 CO_Q3_Mathematics 8_ Module 3 7. In the figure, 𝛥𝐶𝑅𝑌 ≅ 𝛥𝑂𝐵𝑆, what is the side corresponding to 𝑆𝐵 ̅̅̅̅ ? A. 𝐵𝐶 B. 𝐶𝑅 ̅̅̅̅ C. 𝑌𝐶 ̅̅̅̅ D. 𝑌𝑅 ̅̅̅̅ 8. Use the marked triangles in writing a correct congruence statement. A. 𝐶𝑅 ̅̅̅̅ ≅ 𝑇𝑌 ̅̅̅̅ C. 𝑇𝑂 ̅̅̅̅ ≅ 𝐶𝐴 ̅̅̅̅ 𝑂𝑌 ̅̅̅̅ ≅ 𝐶𝐴 ̅̅̅̅ 𝑇𝑌 ̅̅̅̅ ≅ 𝑅𝐶 ̅̅̅̅ 𝑅𝐴 ̅̅̅̅ ≅ 𝑂𝑌 ̅̅̅̅ 𝑂𝑌 ̅̅̅̅ ≅ 𝐴𝐶 ̅̅̅̅ 𝛥𝑇𝑂𝑌 ≅ 𝛥𝐶𝐴𝑅 𝛥𝑇𝑂𝑌 ≅ 𝛥𝐶𝐴𝑅 B. 𝐶𝑅 ̅̅̅̅ ≅ 𝑇𝑌 ̅̅̅̅ D. 𝑂𝑇 ̅̅̅̅ ≅ 𝐴𝐶 ̅̅̅̅ 𝑂𝑌 ̅̅̅̅ ≅ 𝐴𝑅 ̅̅̅̅ 𝑇𝑌 ̅̅̅̅ ≅ 𝐶𝑅 ̅̅̅̅ 𝑇𝑌 ̅̅̅̅ ≅ 𝑅𝐴 ̅̅̅̅ 𝑂𝑌 ̅̅̅̅ ≅ 𝐴𝑅 ̅̅̅̅ 𝛥𝑇𝑂𝑌 ≅ 𝛥𝐶𝐴𝑅 𝛥𝑇𝑂𝑌 ≅ 𝛥𝐶𝐴𝑅 For items 9-12, refer to the figure shown below. 9. Which of the following shows a pair of congruent triangles? A. 𝛥𝐹𝐼𝑉 ≅ 𝛥𝐼𝑉𝐸 C. 𝛥𝐼𝐹𝑉 ≅ 𝛥𝐼𝑉𝐸 B. 𝛥𝐹𝐼𝑉 ≅ 𝛥𝐸𝐼𝑉 D. 𝛥𝐼𝐹𝑉 ≅ 𝛥𝑉𝐼𝐸 𝐶 𝑅 𝑌 𝑂 𝐵 𝑆 𝑇 𝑂 𝑌 𝑅 𝐶 𝐴 𝐼 𝐹 𝑉 𝐸
17 CO_Q3_Mathematics 8_ Module 3 10. What angle corresponds to ∠𝐹𝑉𝐼? A. ∠𝐸 C. ∠𝐸𝐼𝑉 B. ∠𝐹 D. ∠𝐸𝑉𝐼 11. Which segment is congruent to 𝐸𝑉 ̅̅̅̅? A. 𝐼𝐸 ̅̅̅ C. 𝐹𝑉 ̅̅̅̅ B. 𝐼𝑉 ̅̅̅ D. 𝑉𝐸 ̅̅̅̅ 12. Which vertex corresponds to 𝐹? A. 𝐸 C. 𝐹 B. 𝐼 D. 𝑉 13.Which of the following figures show congruent triangles? A. C. B. D. 14.What would be the name of the congruent triangles given their corresponding sides: 𝐿𝑂 ̅̅̅̅ ≅ 𝑅𝑃 ̅̅̅̅, 𝑀𝑂 ̅̅̅̅̅ ≅ 𝑄𝑃 ̅̅̅̅, 𝐿𝑀 ̅̅̅̅ ≅ 𝑅𝑄 ̅̅̅̅? A. 𝛥𝐿𝑂𝑀 ≅ 𝛥𝑄𝑃𝑅 C. 𝛥𝑀𝑂𝐿 ≅ 𝛥𝑅𝑄𝑃 B. 𝛥𝑂𝑀𝐿 ≅ 𝛥𝑃𝑄𝑅 D. 𝛥𝐿𝑀𝑂 ≅ 𝛥𝑃𝑄𝑅 15.Which of the following illustrations below best describe the congruent triangle in item 14? A. C. B. D. 𝐴 𝐷 𝐶 𝐵 𝐸 𝑄 𝑃 𝑅 𝑆 𝑄 𝑇 𝑈 𝑉 𝑊 𝑂 𝐶 𝑂 𝐿 𝑉 𝐸 𝑀 𝑂 𝐿 𝑅 𝑃 𝑄 𝑄 𝐿 𝑂 𝑅 𝑀 𝑃 𝑀 𝐿 𝑂 𝑄 𝑅 𝑃 𝐿 𝑀 𝑂 𝑄 𝑃 𝑅
18 CO_Q3_Mathematics 8_ Module 3 Additional Activities Directions: Do what is asked. Use separate paper to draw your answers. Given:𝛥𝑅𝐸𝐷 ≅ 𝛥𝐸𝑅𝐴 1. Separate the overlapping triangles. 2. Put identical/similar markings to show clearly the congruent sides and congruent angles. 𝑊 𝐷 𝐴 𝑅 𝐸
19 CO_Q3_Mathematics 8_ Module 3 Answer Key What’s New 1. Triangles 2. The triangles have the same size and shape. 3. If we put these triangles one over the other, they would fit exactly or coincide. What’s In For 𝛥𝐴𝐵𝐶 Vertices: A, B, C Angles: ∠𝐴, ∠𝐵, ∠𝐶 Sides: 𝐴𝐵, ̅̅̅̅̅ 𝐴𝐶 ̅̅̅̅ , 𝐵𝐶 ̅̅̅̅ For 𝛥𝑄𝑅𝑆 Vertices: Q, R, S Angles: ∠𝑄, ∠𝑅, ∠𝑆 Sides: 𝑄𝑅, ̅̅̅̅̅ 𝑄𝑆 ̅̅̅̅ , 𝑅𝑆 ̅̅̅̅ Angle - A pair of rays that have a common endpoint, provided that the rays do not lie on the line. Vertex- In an angle, it is the common endpoint of two rays. In a polygon, it is the common endpoint of two sides. Sides – In an angle, the sides are rays that form the angle. In a polygon, the sides are the segments that form a polygon. What I Know 1. D 2. C 3. D 4. D 5. C 6. B 7. C 8. C 9. A 10. C 11. B 12. C 13. D 14. A 15. C 𝐴 ↔ 𝐷 𝐵 ↔ 𝐸 ∠𝐴 ↔ ∠𝐷 ∠𝐵 ↔ ∠𝐸 ∠𝐶 ↔ ∠𝐹 𝐴𝐵 ̅̅̅̅ ↔ 𝐷𝐸 ̅̅̅̅ 𝐴𝐶 ̅̅̅̅ ↔ 𝐷𝐹 ̅̅̅̅ 𝐵𝐶 ̅̅̅̅ ↔ 𝐸𝐹 ̅̅̅̅ ∠𝑆 ≅ ∠𝑅 ∠𝑇 ≅ ∠𝐴 ∠𝑅 ≅ ∠𝑆 𝑆𝑇 ̅̅ ̅̅ ≅ 𝑅𝐴 ̅̅̅̅ 𝑇𝑅 ̅̅̅̅ ≅ 𝐴𝑆 ̅̅̅̅ 𝑆𝑅 ̅̅̅̅ ≅ 𝑅𝑆 ̅̅̅̅ What’s More 1. Corresponding Vertices Corresponding Angles Corresponding Sides 2. Note: Illustration varies however the congruent parts must be considered. 3. Congruent Angles Congruent Sides 4. 𝛥𝑇𝑂𝑊 ≅ 𝛥𝐸𝑂𝑁 Assessment 1. A 2. D 3. C 4. B 5. C 6. D 7. D 8. D 9. B 10. D 11. C 12. A 13. D 14. B 15. D ∠𝐴 ≅ ∠𝐴 ∠𝐹 ≅ ∠𝐹 ∠𝑆 ≅ ∠𝐸 𝐴𝑆 ̅̅̅̅ ≅ 𝐴𝐸 ̅̅̅̅ 𝑆𝐹 ̅̅̅̅ ≅ 𝐸𝐹 ̅̅̅̅ 𝐴𝐹 ̅̅̅̅ ≅ 𝐴𝐹 ̅̅̅̅ What I Can Do Congruent Angles Congruent Sides 𝐶 ↔ 𝑅 𝑂 ↔ 𝐸 ∠𝐶 ↔ ∠𝑅 ∠𝑂 ↔ ∠𝐸 𝐶𝑂 ̅̅̅̅ ↔ 𝑅𝐸 ̅̅̅̅ 𝐶𝑀 ̅̅̅̅̅ ↔ 𝑅𝑀 ̅̅̅̅̅ ∠𝐶 ≅ ∠𝑅 ∠𝑂 ≅ ∠𝐸 𝐶𝑂 ̅̅̅̅ ≅ 𝑅𝐸 ̅̅̅̅ 𝐶𝑀 ̅̅̅̅̅ ≅ 𝑅𝑀 ̅̅̅̅̅ What I Have Learned 1. 𝛥𝑀𝐸𝑅 ≅ 𝛥𝑀𝑂𝐶 2. 𝑀 ↔ 𝑀 3. ∠𝑀 ↔ ∠𝑀 4. 𝑀𝑂 ̅̅̅̅̅ ↔ 𝑀𝐸 ̅̅̅̅̅ 5. ∠𝑀 ≅ ∠𝑀 6. 𝑀𝑂 ̅̅̅̅̅ ≅ 𝑀𝐸 ̅̅̅̅̅ Additional Activities
20 CO_Q3_Mathematics 8_ Module 3 References Abuzo, Emmanuel P., Bryant, Merden L., Cabrella, Jem Boy B. Caldez, Belen P., Callanta, Melvin M., Castro, Anastacia Preserfina I., Halabaso, Alicia R., Javier, Sonia P., Nocom, Roger T., and Ternida, Conception S. (2013). Grade 8 Mathematics Learners Module. pp. 352-353. Philippines. Book Media Press, Inc. and Printwell, Inc. Bernabe, Julieta G., et.al. (2009). Revised Edition Geometry, pp. 88-90, Gregoria Araneta Avenue, Quezon City, Philippines. SD Publications, Inc. Orines, Fernando B., et.al. (2008). Second Edition Next Century Mathematics, pp. 201-203, Quezon Avenue, Quezon City, Philippines. Phoenix Publishing House, Inc.
For inquiries or feedback, please write or call: Department of Education - Bureau of Learning Resources (DepEd-BLR) Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600 Telefax: (632) 8634-1072; 8634-1054; 8631-4985 Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph

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    CO_Q3_Mathematics 8_ Module3 Mathematics Quarter 3 – Module 3 Illustrating Triangle Congruence
  • 2.
    Mathematics – Grade8 Alternative Delivery Mode Quarter 3 – Module 3: Illustrating Triangle Congruence 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 Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio Printed in the Philippines __________________________ Department of Education – Caraga Region Office Address: Learning Resource Management Section (LRMS)tr555555 J.P. Rosales Avenue, Butuan City, Philippines 8600 Telefax Nos.: (085) 342-8207 / (085) 342-5969 E-mail Address: caraga@deped.gov.ph Development Team of the Module Writer: Rochelle Anna Myla A. Paronda, Language Editor: Joel B. Asonto Content Editors: Victorino Nimes, Myracell Buenaflor, Marie Grace O. Aparre, Ella C. Armayan Reviewers: Rhea J. Yparraguirre, Villaflor D. Edillor, Shirley C. Compe, Imelda B. Llasos, Caren L. Alamin, Jayrold Virtudazo Navarra, Alma R. Velaasco Illustrator: Fritch A. Paronda Layout Artists: Fritch A. Paronda, James B. De Los Reyes Management Team: Francis Cesar B. Bringas Isidro M. Biol, Jr. Maripaz F. Magno Josephine Chonie M. Obseñares Josita B. Carmen Celsa A. Casa Regina Euann A. Puerto Bryan L. Arreo Elnie Anthony P. Barcena Leopardo P. Cortes, Jr. Claire Ann P. Gonzaga
  • 3.
    8 Mathematics Quarter 3 –Module 3 Illustrating Triangle Congruence
  • 4.
    Introductory Message This Self-LearningModule (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson. Each SLM is composed of different parts. Each part shall guide you step-by- step as you discover and understand the lesson prepared for you. Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self- check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these. In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning. Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task. If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Thank you.
  • 5.
    1 CO_Q3_Mathematics 8_ Module3 What I Need to Know This module was designed and written with you in mind. It is here to help you illustrate triangle congruence. You are provided with varied activities to process the knowledge and skills learned and to deepen and transfer your understanding of the lesson. The scope of this module enables you to use it in many different learning situations. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using. This module contains concepts on how to illustrate triangle congruence. After going through this module, you are expected to: 1. define triangle congruence; 2. draw and label the corresponding parts of two congruent triangles; 3. identify corresponding parts of two congruent triangles; and 3. relate triangle congruence in real-life.
  • 6.
    2 CO_Q3_Mathematics 8_ Module3 What I Know Directions: Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper. 1. Which of the following is a symbol used for congruence? A. = C. ≈ B. ≠ D. ≅ 2. How many pairs of corresponding congruent angles are there in two congruent triangles? A. 1 C. 3 B. 2 D. 4 3. Which of the symbols below is used to indicate correspondence? A. ≅ C. 𝛥 B. ∠ D. ↔ 4. Which of the following statements best describe congruent triangles? I. They are similar. II. They have the same size. III. They have the same shape. IV. They are equilateral triangles. A. I only C. III and IV together B. III only D. II and III together 5. How many pairs of corresponding sides are there in two congruent triangles? A. 1 C. 3 B. 2 D. 4 6. Which of the following statements best describe the corresponding parts of congruent triangles? A. They are not equal. B. They are congruent. C. They are supplementary. D. They are complementary.
  • 7.
    3 CO_Q3_Mathematics 8_ Module3 For items 7 - 10, consider the figure below. 7. Which segment corresponds to 𝑁𝑇 ̅̅̅̅ ? A. 𝐴𝑇 ̅̅̅̅ C. 𝐶𝑀 ̅̅̅̅̅ B. 𝐴𝐶 ̅̅̅̅ D. 𝑀𝐴 ̅̅̅̅̅ 8. Which angle corresponds to ∠𝑇 ? A. ∠𝐴 C. ∠𝑀 B. ∠𝐶 D. ∠𝑁 9. Which vertex corresponds to 𝐴? A. 𝐴 C. 𝑁 B. 𝑀 D. 𝑇 10.What conclusion can you make in the figure? A. 𝛥𝐴𝑁𝑇 ≅ 𝛥𝑀𝐴𝐶 B. 𝛥𝐴𝑁𝑇 ≅ 𝛥𝐶𝐴𝑀 C. 𝛥𝐴𝑀𝐶 ≅ 𝛥𝐴𝑇𝑁 D. 𝛥𝐴𝑀𝐶 ≅ 𝛥𝑇𝐴𝑁 11.If 𝛥𝐿𝑀𝑁 ≅ 𝛥𝑃𝑅𝑄, then what angle corresponds to ∠𝑄 ? A. ∠𝐿 C. ∠𝑃 B. ∠𝑁 D. ∠𝑅 12.Which of the following figures illustrate pair of congruent triangles? A. C. B. D. 𝑀 𝑁 𝐶 𝑇 𝐴 A D C B E O P R S T Q U V W A B C D
  • 8.
    4 CO_Q3_Mathematics 8_ Module3 13.Which of the following correspondence is equivalent to ∠𝐽𝐾𝐿 ≅ ∠𝑀𝑁𝑂? A. ∠𝐾𝐽𝐿 ↔ ∠𝑀𝑂𝑁 B. ∠𝐾𝐽𝐿 ↔ ∠𝑂𝑀𝑁 C. ∠𝐽𝐿𝐾 ↔ ∠𝑁𝑀𝑂 D. ∠𝐿𝐾𝐽 ↔ ∠𝑂𝑁𝑀 14. If 𝛥𝑁𝐴𝑇 ≅ 𝛥𝐺𝐸𝑂, then name the corresponding sides of the congruent triangles. A. 𝑁𝐴 ̅̅̅̅ ≅ 𝐺𝐸 ̅̅̅̅ C. 𝐴𝑇 ̅̅̅̅ ≅ 𝐸𝑂 ̅̅̅̅ 𝐴𝑇 ̅̅̅̅ ≅ 𝐸𝑂 ̅̅̅̅ 𝑁𝑇 ̅̅̅̅ ≅ 𝐸𝐺 ̅̅̅̅ 𝑁𝑇 ̅̅̅̅ ≅ 𝐺𝑂 ̅̅̅̅ 𝑁𝐴 ̅̅̅̅ ≅ 𝑂𝐺 ̅̅̅̅ B. 𝑁𝐴 ̅̅̅̅ ≅ 𝐺𝑂 ̅̅̅̅ D. 𝐴𝑇 ̅̅̅̅ ≅ 𝐺𝐸 ̅̅̅̅ 𝐴𝑇 ̅̅̅̅ ≅ 𝐸𝑂 ̅̅̅̅ 𝑁𝐴 ̅̅̅̅ ≅ 𝐸𝑂 ̅̅̅̅ 𝑁𝑇 ̅̅̅̅ ≅ 𝐺𝐸 ̅̅̅̅ 𝑁𝑇 ̅̅̅̅ ≅ 𝐺𝑂 ̅̅̅̅ 15.Lyka was asked by her Math teacher to give and write the corresponding sides of congruent triangles, 𝛥𝐶𝐴𝑇 and 𝛥𝐷𝑂𝐺, on the board. Her answer is shown below. 𝐴𝐶 ̅̅̅̅ ↔ 𝑂𝐷 ̅̅̅̅ 𝐶𝑇 ̅̅̅̅ ↔ 𝐷𝐺 ̅̅̅̅ 𝐴𝑇 ̅̅̅̅ ↔ 𝑂𝐷 ̅̅̅̅ Is Lyka’s answer correct? A. Yes, because each side of 𝛥𝐶𝐴𝑇 is paired correctly to each side of 𝛥𝐷𝑂𝐺. B. Yes, because each angle of 𝛥𝐶𝐴𝑇 is paired correctly to each angle of 𝛥𝐷𝑂𝐺. C. No, because 𝐴𝑇 ̅̅̅̅ corresponds to 𝑂𝐺 ̅̅̅̅. D. No, because 𝐶𝑇 ̅̅̅̅ corresponds to 𝑂𝐺 ̅̅̅̅.
  • 9.
    5 CO_Q3_Mathematics 8_ Module3 Lesson 1 Illustrating Triangle Congruence Designs and patterns having the same size and shape thrive in almost every place. These can be seen in bridges and condominiums, furniture and appliances, as well as in fabrics and crafts. Triangle is one of the most important figures studied in Mathematics because of its many uses in the real-world. Let us know more about triangles, particularly congruent triangles by performing the array of activities in this module. Have fun! What’s In Activity: Identify V-A-S (Vertices – Angles – Sides) Directions: Identify the vertices, angles, and sides of each triangle. Then, answer the questions below using complete sentences. Questions: 1. How did you identify the vertices, angles, and sides of each triangle? 2. How would you define angle? 3. What about the definition of a vertex? 4. In geometry, what is meant by side? Vertices Angles Sides Vertices Angles Sides 𝐵 𝐴 𝐶 𝑅 𝑄 𝑆
  • 10.
    6 CO_Q3_Mathematics 8_ Module3 What’s New Activity: Four Pics – Two Words Directions: Consider the figures below and answer the questions that follow. Questions 1. What geometric figures were highlighted in the illustrations? 2. What have you observed with the size and shape of the geometric figures? Explain your answer. What is It  Two line segments are congruent if their endpoints can be made to coincide.  Two angles are congruent if their sides can be made to coincide.  Two triangles are equal if they have the same size and shape.  Two triangles are congruent if their corresponding parts are congruent.
  • 11.
    7 CO_Q3_Mathematics 8_ Module3 Assume 𝛥𝑁𝐴𝑇 coincides with 𝛥𝐺𝐸𝑂, such that the vertices of 𝛥𝑁𝐴𝑇 fit exactly over the vertices of 𝛥𝐺𝐸𝑂 . The correspondence can be created between triangles. 𝛥𝑁𝐴𝑇 ↔ 𝛥𝐺𝐸𝑂 In this correspondence, there are three pairs of corresponding vertices, angles, and sides. Since the vertices of 𝛥𝑁𝐴𝑇 fit exactly with the vertices of 𝛥𝐺𝐸𝑂, therefore 𝛥𝑁𝐴𝑇 and 𝛥𝐺𝐸𝑂 are congruent. In symbols, 𝛥𝑁𝐴𝑇 ≅ 𝛥𝐺𝐸𝑂 Definition of Correspondence Pairing the parts of one group to the parts of another group is called correspondence. Correspondence uses the notation “↔”. Corresponding Angles ∠𝑁 ↔ ∠𝐺 ∠𝐴 ↔ ∠𝐸 ∠𝑇 ↔ ∠𝑂 𝐴 𝑁 𝑇 𝐸 𝐺 𝑂 Corresponding Sides 𝑁𝐴 ̅̅̅̅ ↔ 𝐺𝐸 ̅̅̅̅ 𝐴𝑇 ̅̅̅̅ ↔ 𝐸𝑂 ̅̅̅̅ 𝑁𝑇 ̅̅̅̅ ↔ 𝐺𝑂 ̅̅̅̅ Corresponding Vertices 𝑁 ↔ 𝐺 𝐴 ↔ 𝐸 𝑇 ↔ 𝑂 Note In naming congruent triangles, the order of the letters representing the vertices of one triangle shall be written in the same order as their corresponding vertices in the other triangle. Definition of Congruent Triangles Two triangles are congruent if and only if the corresponding sides and the corresponding angles are congruent.
  • 12.
    8 CO_Q3_Mathematics 8_ Module3 (𝛥𝑁𝐴𝑇 ≅ 𝛥𝐺𝐸𝑂 is read as “triangle 𝑁 − 𝐴 − 𝑇 is congruent to triangle 𝐺 − 𝐸 − 𝑂”) It implies that the corresponding parts of congruent triangles are also congruent. Conversely, if the corresponding parts of the two triangles are congruent, then the two triangles are congruent. Example 1 𝛥𝐶𝐴𝑇 ≅ 𝛥𝐷𝑂𝐺. Write down the pairs of corresponding parts. Solution The congruent parts of congruent triangles can be determined with the help of arcs and tick-marks. CPCTC Corresponding Parts of Congruent Triangles are Congruent. Congruent Angles ∠𝑁 ≅ ∠𝐺 ∠𝐴 ≅ ∠𝐸 ∠𝑇 ≅ ∠𝑂 Congruent Sides 𝑁𝐴 ̅̅̅̅ ≅ 𝐺𝐸 ̅̅̅̅ 𝐴𝑇 ̅̅̅̅ ≅ 𝐸𝑂 ̅̅̅̅ 𝑁𝑇 ̅̅̅̅ ≅ 𝐺𝑂 ̅̅̅̅ Corresponding Sides 𝐶𝐴 ̅̅̅̅ ↔ 𝐷𝑂 ̅̅̅̅ 𝐶𝑇 ̅̅̅̅ ↔ 𝐷𝐺 ̅̅̅̅ 𝐴𝑇 ̅̅̅̅ ↔ 𝑂𝐺 ̅̅̅̅ Corresponding Angles ∠𝐶 ↔ ∠𝐷 ∠𝐴 ↔ ∠𝑂 ∠𝑇 ↔ ∠𝐺 𝐴 𝐶 𝑇 𝑂 𝐷 𝐺
  • 13.
    9 CO_Q3_Mathematics 8_ Module3 Example 2 Given that the two triangles below are congruent, list the pairs of congruent angles and congruent sides and then name the congruent triangles. Solution 𝛥𝑃𝑆𝐴 ≅ 𝛥𝑁𝑆𝑂 Example 3 Consider 𝛥𝐹𝐴𝐸 ≅ 𝛥𝑇𝐸𝐴. List down the pairs of congruent angles and congruent sides. Solution Again, the congruent parts of congruent triangles can be determined with the help of arcs and tick-marks. Since 𝛥𝐹𝐴𝐸 ≅ 𝛥𝑇𝐸𝐴, then the corresponding parts are as follows: 𝐸 𝑇 𝐹 𝐴 Corresponding Angles ∠𝐹 ↔ ∠𝑇 ∠𝐹𝐸𝐴 ↔ ∠𝑇𝐴𝐸 ∠𝐹𝐴𝐸 ↔ ∠𝑇𝐸𝐴 Corresponding Sides 𝐹𝐴 ̅̅̅̅ ↔ 𝑇𝐸 ̅̅̅̅ 𝐹𝐸 ̅̅̅̅ ↔ 𝑇𝐴 ̅̅̅̅ 𝐸𝐴 ̅̅̅̅ ↔ 𝐴𝐸 ̅̅̅̅ 𝑃 𝐴 𝑆 𝑁 𝑂 Congruent Angles ∠𝐴 ≅ ∠𝑂 ∠𝑃 ≅ ∠𝑁 ∠𝑃𝑆𝐴 ≅ ∠𝑁𝑆𝑂 Congruent Sides 𝑃𝑆 ̅̅̅̅ ≅ 𝑁𝑆 ̅̅̅̅ 𝑃𝐴 ̅̅̅̅ ≅ 𝑁𝑂 ̅̅̅̅ 𝑆𝐴 ̅̅̅̅ ≅ 𝑆𝑂 ̅̅̅̅
  • 14.
    10 CO_Q3_Mathematics 8_ Module3 Example 4 Assume that segments and angles in the numbered triangles are congruent. Identify which triangles are congruent. Solution Looking at the figures, it can be deduced that 𝛥1 ≅ 𝛥6, 𝛥2 ≅ 𝛥5, and 𝛥3 ≅ 𝛥4. Remember:  Triangles that have exactly the same size and shape are congruent.  Two triangles are congruent if their corresponding sides and angles are congruent.  Two triangles are congruent if they fit exactly or coincide when one triangle is put on top of the other. 1 2 3 4 5 6
  • 15.
    11 CO_Q3_Mathematics 8_ Module3 What’s More Activity 1: Find My PARTner Description: In this activity, you will be pairing parts of one triangle to the parts of another triangle. Note that the notation “↔” shows corresponding parts. Directions: Copy the table on your answer sheet and supply the missing part of the triangle that will make each correspondence complete. Answer the questions that follow. Corresponding Vertices Corresponding Angles Corresponding Sides 𝐴 ↔ _____ _____ ↔ 𝐸 𝐶 ↔ 𝐹 ∠𝐴 ↔ _____ ∠𝐵 ↔ _____ ∠𝐶 ↔ _____ 𝐴𝐵 ̅̅̅̅ ↔ _____ _____ ↔ 𝐷𝐹 ̅̅̅̅ 𝐵𝐶 ̅̅̅̅ ↔ _____ Questions: 1. What have you noticed on the given triangles? 2. What does the mark of the angles and sides of the triangle indicate? 3. Are there angles and sides that have the same measure and length? Explain your answer. 4. Describe the relationship between 𝛥𝐴𝐵𝐶 and𝛥𝐷𝐸𝐹. 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹
  • 16.
    12 CO_Q3_Mathematics 8_ Module3 Activity 2: Draw and Name Directions: Do what is asked. Use a separate sheet of paper. 1. Draw 𝛥𝐴𝐶𝑇 ≅ 𝛥𝑁𝑂𝑊. Put identical markings on corresponding congruent parts. 2. 𝛥𝑆𝑇𝑅 ≅ 𝛥𝑅𝐴𝑆. Name the six pairs of corresponding congruent parts. 3. Listed below are the six pairs of corresponding parts of two congruent triangles. Name the two congruent triangles. Congruent Angles Congruent Sides ∠𝑇 ≅ ∠𝐸 ∠𝑂 ≅ ∠𝑂 ∠𝑊 ≅ ∠𝑁 𝑇𝑊 ̅̅̅̅̅ ≅ 𝐸𝑁 ̅̅̅̅ 𝑂𝑇 ̅̅̅̅ ≅ 𝑂𝐸 ̅̅̅̅ 𝑊𝑂 ̅̅̅̅̅ ≅ 𝑁𝑂 ̅̅̅̅ 𝑇 𝐴 𝑆 𝑅 Congruent Angles Congruent Sides Congruent Triangles
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    13 CO_Q3_Mathematics 8_ Module3 What I Have Learned Let us see how far you have learned about congruent triangles by answering each question in the diagram below. Consider the figure at the center. Write your answers on a separate paper. What are the congruent triangles? What are the corresponding vertices? What are the corresponding angles? What are the corresponding sides? What are the congruent angles? What are the congruent sides? 𝐸 𝑅 𝐶 𝑂 𝑀
  • 18.
    14 CO_Q3_Mathematics 8_ Module3 What I Can Do One of the structural applications of triangles can be seen in bridges. In the illustration below, there are pairs of congruent triangles. Do what is asked below and write your answer on a separate paper. Directions: 1. Select a pair of congruent triangles. 2. Label these triangles as 𝛥𝑆𝐴𝐹 and 𝛥𝐸𝐴𝐹. 3. Put similar markings to show clearly the congruent sides and congruent angles. 4. List down the corresponding congruent parts of the congruent triangles.
  • 19.
    15 CO_Q3_Mathematics 8_ Module3 Assessment Directions: Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper. 1. Which of the following statements is not true about congruent triangles? A. They are dissimilar. B. They are coinciding. C. They have the same size and shape. D. They have corresponding congruent parts. 2. How many pairs of corresponding congruent parts are there in two congruent triangles? A. 2 C. 4 B. 3 D. 6 3. If 𝛥𝐹𝑅𝑌 ≅ 𝛥𝐻𝑂𝑇, which segment is congruent to 𝑅𝑌 ̅̅̅̅? A. 𝐻𝑇 ̅̅̅̅ C. 𝑂𝑇 ̅̅̅̅ B. 𝐹𝑌 ̅̅̅̅ D. 𝑅𝐹 ̅̅̅̅ 4. Which of the following is true about the corresponding parts of congruent triangles? A. They are unequal. B. They are congruent. C. They are supplementary. D. They are complementary. 5. If 𝛥𝑆𝐼𝑇 ≅ 𝛥𝐻𝑂𝑃, then what angle corresponds to ∠𝑇? A. ∠𝐻 C. ∠𝑃 B. ∠𝐼 D. ∠𝑆 6. Name the congruent triangles whose pairs of corresponding parts are shown below. ∠𝐻 ≅ ∠𝑆 𝑅𝐸 ̅̅̅̅ ≅ 𝑁𝑂 ̅̅̅̅ ∠𝑅 ≅ ∠𝑁 𝐻𝐸 ̅̅̅̅ ≅ 𝑆𝑂 ̅̅̅̅ ∠𝐸 ≅ ∠𝑂 𝐻𝑅 ̅̅̅̅ ≅ 𝑆𝑁 ̅̅̅̅ A. 𝛥𝑁𝑂𝑆 ≅ 𝛥𝐻𝐸𝑅 C. 𝛥𝑆𝑂𝑁 ≅ 𝛥𝐸𝐻𝑅 B. 𝛥𝑁𝑆𝑂 ≅ 𝛥𝐸𝑅𝐻 D. 𝛥𝑆𝑂𝑁 ≅ 𝛥𝐻𝐸𝑅
  • 20.
    16 CO_Q3_Mathematics 8_ Module3 7. In the figure, 𝛥𝐶𝑅𝑌 ≅ 𝛥𝑂𝐵𝑆, what is the side corresponding to 𝑆𝐵 ̅̅̅̅ ? A. 𝐵𝐶 B. 𝐶𝑅 ̅̅̅̅ C. 𝑌𝐶 ̅̅̅̅ D. 𝑌𝑅 ̅̅̅̅ 8. Use the marked triangles in writing a correct congruence statement. A. 𝐶𝑅 ̅̅̅̅ ≅ 𝑇𝑌 ̅̅̅̅ C. 𝑇𝑂 ̅̅̅̅ ≅ 𝐶𝐴 ̅̅̅̅ 𝑂𝑌 ̅̅̅̅ ≅ 𝐶𝐴 ̅̅̅̅ 𝑇𝑌 ̅̅̅̅ ≅ 𝑅𝐶 ̅̅̅̅ 𝑅𝐴 ̅̅̅̅ ≅ 𝑂𝑌 ̅̅̅̅ 𝑂𝑌 ̅̅̅̅ ≅ 𝐴𝐶 ̅̅̅̅ 𝛥𝑇𝑂𝑌 ≅ 𝛥𝐶𝐴𝑅 𝛥𝑇𝑂𝑌 ≅ 𝛥𝐶𝐴𝑅 B. 𝐶𝑅 ̅̅̅̅ ≅ 𝑇𝑌 ̅̅̅̅ D. 𝑂𝑇 ̅̅̅̅ ≅ 𝐴𝐶 ̅̅̅̅ 𝑂𝑌 ̅̅̅̅ ≅ 𝐴𝑅 ̅̅̅̅ 𝑇𝑌 ̅̅̅̅ ≅ 𝐶𝑅 ̅̅̅̅ 𝑇𝑌 ̅̅̅̅ ≅ 𝑅𝐴 ̅̅̅̅ 𝑂𝑌 ̅̅̅̅ ≅ 𝐴𝑅 ̅̅̅̅ 𝛥𝑇𝑂𝑌 ≅ 𝛥𝐶𝐴𝑅 𝛥𝑇𝑂𝑌 ≅ 𝛥𝐶𝐴𝑅 For items 9-12, refer to the figure shown below. 9. Which of the following shows a pair of congruent triangles? A. 𝛥𝐹𝐼𝑉 ≅ 𝛥𝐼𝑉𝐸 C. 𝛥𝐼𝐹𝑉 ≅ 𝛥𝐼𝑉𝐸 B. 𝛥𝐹𝐼𝑉 ≅ 𝛥𝐸𝐼𝑉 D. 𝛥𝐼𝐹𝑉 ≅ 𝛥𝑉𝐼𝐸 𝐶 𝑅 𝑌 𝑂 𝐵 𝑆 𝑇 𝑂 𝑌 𝑅 𝐶 𝐴 𝐼 𝐹 𝑉 𝐸
  • 21.
    17 CO_Q3_Mathematics 8_ Module3 10. What angle corresponds to ∠𝐹𝑉𝐼? A. ∠𝐸 C. ∠𝐸𝐼𝑉 B. ∠𝐹 D. ∠𝐸𝑉𝐼 11. Which segment is congruent to 𝐸𝑉 ̅̅̅̅? A. 𝐼𝐸 ̅̅̅ C. 𝐹𝑉 ̅̅̅̅ B. 𝐼𝑉 ̅̅̅ D. 𝑉𝐸 ̅̅̅̅ 12. Which vertex corresponds to 𝐹? A. 𝐸 C. 𝐹 B. 𝐼 D. 𝑉 13.Which of the following figures show congruent triangles? A. C. B. D. 14.What would be the name of the congruent triangles given their corresponding sides: 𝐿𝑂 ̅̅̅̅ ≅ 𝑅𝑃 ̅̅̅̅, 𝑀𝑂 ̅̅̅̅̅ ≅ 𝑄𝑃 ̅̅̅̅, 𝐿𝑀 ̅̅̅̅ ≅ 𝑅𝑄 ̅̅̅̅? A. 𝛥𝐿𝑂𝑀 ≅ 𝛥𝑄𝑃𝑅 C. 𝛥𝑀𝑂𝐿 ≅ 𝛥𝑅𝑄𝑃 B. 𝛥𝑂𝑀𝐿 ≅ 𝛥𝑃𝑄𝑅 D. 𝛥𝐿𝑀𝑂 ≅ 𝛥𝑃𝑄𝑅 15.Which of the following illustrations below best describe the congruent triangle in item 14? A. C. B. D. 𝐴 𝐷 𝐶 𝐵 𝐸 𝑄 𝑃 𝑅 𝑆 𝑄 𝑇 𝑈 𝑉 𝑊 𝑂 𝐶 𝑂 𝐿 𝑉 𝐸 𝑀 𝑂 𝐿 𝑅 𝑃 𝑄 𝑄 𝐿 𝑂 𝑅 𝑀 𝑃 𝑀 𝐿 𝑂 𝑄 𝑅 𝑃 𝐿 𝑀 𝑂 𝑄 𝑃 𝑅
  • 22.
    18 CO_Q3_Mathematics 8_ Module3 Additional Activities Directions: Do what is asked. Use separate paper to draw your answers. Given:𝛥𝑅𝐸𝐷 ≅ 𝛥𝐸𝑅𝐴 1. Separate the overlapping triangles. 2. Put identical/similar markings to show clearly the congruent sides and congruent angles. 𝑊 𝐷 𝐴 𝑅 𝐸
  • 23.
    19 CO_Q3_Mathematics 8_ Module3 Answer Key What’s New 1. Triangles 2. The triangles have the same size and shape. 3. If we put these triangles one over the other, they would fit exactly or coincide. What’s In For 𝛥𝐴𝐵𝐶 Vertices: A, B, C Angles: ∠𝐴, ∠𝐵, ∠𝐶 Sides: 𝐴𝐵, ̅̅̅̅̅ 𝐴𝐶 ̅̅̅̅ , 𝐵𝐶 ̅̅̅̅ For 𝛥𝑄𝑅𝑆 Vertices: Q, R, S Angles: ∠𝑄, ∠𝑅, ∠𝑆 Sides: 𝑄𝑅, ̅̅̅̅̅ 𝑄𝑆 ̅̅̅̅ , 𝑅𝑆 ̅̅̅̅ Angle - A pair of rays that have a common endpoint, provided that the rays do not lie on the line. Vertex- In an angle, it is the common endpoint of two rays. In a polygon, it is the common endpoint of two sides. Sides – In an angle, the sides are rays that form the angle. In a polygon, the sides are the segments that form a polygon. What I Know 1. D 2. C 3. D 4. D 5. C 6. B 7. C 8. C 9. A 10. C 11. B 12. C 13. D 14. A 15. C 𝐴 ↔ 𝐷 𝐵 ↔ 𝐸 ∠𝐴 ↔ ∠𝐷 ∠𝐵 ↔ ∠𝐸 ∠𝐶 ↔ ∠𝐹 𝐴𝐵 ̅̅̅̅ ↔ 𝐷𝐸 ̅̅̅̅ 𝐴𝐶 ̅̅̅̅ ↔ 𝐷𝐹 ̅̅̅̅ 𝐵𝐶 ̅̅̅̅ ↔ 𝐸𝐹 ̅̅̅̅ ∠𝑆 ≅ ∠𝑅 ∠𝑇 ≅ ∠𝐴 ∠𝑅 ≅ ∠𝑆 𝑆𝑇 ̅̅ ̅̅ ≅ 𝑅𝐴 ̅̅̅̅ 𝑇𝑅 ̅̅̅̅ ≅ 𝐴𝑆 ̅̅̅̅ 𝑆𝑅 ̅̅̅̅ ≅ 𝑅𝑆 ̅̅̅̅ What’s More 1. Corresponding Vertices Corresponding Angles Corresponding Sides 2. Note: Illustration varies however the congruent parts must be considered. 3. Congruent Angles Congruent Sides 4. 𝛥𝑇𝑂𝑊 ≅ 𝛥𝐸𝑂𝑁 Assessment 1. A 2. D 3. C 4. B 5. C 6. D 7. D 8. D 9. B 10. D 11. C 12. A 13. D 14. B 15. D ∠𝐴 ≅ ∠𝐴 ∠𝐹 ≅ ∠𝐹 ∠𝑆 ≅ ∠𝐸 𝐴𝑆 ̅̅̅̅ ≅ 𝐴𝐸 ̅̅̅̅ 𝑆𝐹 ̅̅̅̅ ≅ 𝐸𝐹 ̅̅̅̅ 𝐴𝐹 ̅̅̅̅ ≅ 𝐴𝐹 ̅̅̅̅ What I Can Do Congruent Angles Congruent Sides 𝐶 ↔ 𝑅 𝑂 ↔ 𝐸 ∠𝐶 ↔ ∠𝑅 ∠𝑂 ↔ ∠𝐸 𝐶𝑂 ̅̅̅̅ ↔ 𝑅𝐸 ̅̅̅̅ 𝐶𝑀 ̅̅̅̅̅ ↔ 𝑅𝑀 ̅̅̅̅̅ ∠𝐶 ≅ ∠𝑅 ∠𝑂 ≅ ∠𝐸 𝐶𝑂 ̅̅̅̅ ≅ 𝑅𝐸 ̅̅̅̅ 𝐶𝑀 ̅̅̅̅̅ ≅ 𝑅𝑀 ̅̅̅̅̅ What I Have Learned 1. 𝛥𝑀𝐸𝑅 ≅ 𝛥𝑀𝑂𝐶 2. 𝑀 ↔ 𝑀 3. ∠𝑀 ↔ ∠𝑀 4. 𝑀𝑂 ̅̅̅̅̅ ↔ 𝑀𝐸 ̅̅̅̅̅ 5. ∠𝑀 ≅ ∠𝑀 6. 𝑀𝑂 ̅̅̅̅̅ ≅ 𝑀𝐸 ̅̅̅̅̅ Additional Activities
  • 24.
    20 CO_Q3_Mathematics 8_ Module3 References Abuzo, Emmanuel P., Bryant, Merden L., Cabrella, Jem Boy B. Caldez, Belen P., Callanta, Melvin M., Castro, Anastacia Preserfina I., Halabaso, Alicia R., Javier, Sonia P., Nocom, Roger T., and Ternida, Conception S. (2013). Grade 8 Mathematics Learners Module. pp. 352-353. Philippines. Book Media Press, Inc. and Printwell, Inc. Bernabe, Julieta G., et.al. (2009). Revised Edition Geometry, pp. 88-90, Gregoria Araneta Avenue, Quezon City, Philippines. SD Publications, Inc. Orines, Fernando B., et.al. (2008). Second Edition Next Century Mathematics, pp. 201-203, Quezon Avenue, Quezon City, Philippines. Phoenix Publishing House, Inc.
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