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10, Mathematics, Quarter 2 – Module 4:, Proving Theorems Related to, Chords, Arcs, Central Angles,, and Inscribed Angles, , CO_Q2_Mathematics 10_ Module 4

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Mathematics – Grade 10, Alternative Delivery Mode, Quarter 2 – Module 4: Proving Theorems Related to Chords, Arcs, Central Angles, and, Inscribed Angles, 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, Development Team of the Module, Author: John Denver B. Pinkihan, Editor’s Name: Aiza R. Bitanga, Reviewer’s Name: Bryan A. Hidalgo, RO EPS for Mathematics, Management Team:, May B. Eclar, Benedicta B. Gamatero, Carmel F. Meris, Ethielyn E. Taqued, Edgar H. Madlaing, Marciana M. Aydinan, Lydia I. Belingon, , Printed in the Philippines by:, Department of Education – Cordillera Administrative Region, Office Address:, Telefax:, E-mail Address:, , Wangal, La Trinidad, Benguet, (074) 422-4074, [email protected]

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10, , Mathematics, Quarter 2 – Module 4:, Proving Theorems Related to, Chords, Arcs, Central Angles,, and Inscribed Angles

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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-bystep 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.

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What I Need to Know, This module was designed and written with you in mind. It is here to help you, prove theorems related to chords, arcs, central angles, and inscribed angles. The scope, of this module permits it to be used in many different learning situations. The language, used recognizes the diverse vocabulary level of students. The lessons are arranged to, follow the standard sequence of the course but the order in which you read and answer, this module is dependent on your ability., This module contains Lesson 1: Theorems Related to Chords, Arcs, and Central, Angles and Lesson 2: Theorems Related to Chords, Arcs, and Inscribed Angles. After, going through this module, you are expected to prove theorems related to chords, arcs,, central angles, and inscribed angles., , What I Know, Directions: Read and analyze each item very carefully. On your answer sheet, write, the letter of the choice that corresponds to the correct answer., 1. Which of the following illustrations do NOT show congruence?, a., c., 𝑟 = 8.25 𝑖𝑛, , 𝑟 = 8.25 𝑖𝑛, , The two circles are congruent., , b., , The two circles are congruent., , d., , 𝐴, , 90°, , 𝐵, , 𝑑 = 14.50 𝑖𝑛, , 𝑟 = 7.25 𝑖𝑛, , 𝐷, , 𝐹, , 𝐸, , 𝐻, , 𝐶, 𝐺, , ̂ ≅ 𝐶𝐷, ̂, 𝐴𝐵, , ̂ ≅ 𝐹𝐻, ̂, 𝐸𝐹, , 2. An inscribed angle is a right angle if it intercepts a __________., a. whole circle, b. semicircle, c. minor arc, d. major arc, 1, , CO_Q2_Mathematics 10_ Module 4

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̂ is a semicircle and m∠𝑆𝐴𝐷 =, 10. In ⊙ 𝐴, what is the measure of ∠𝑆𝐴𝑌 if 𝐷𝑆𝑌, 50?, a. 130°, b. 110°, , c. 100°, d. 50°, , 11. Quadrilateral 𝑆𝑀𝐼𝐿 is inscribed in ⊙ 𝐸., If m∠𝑆𝑀𝐼 = 78 and m∠𝑀𝑆𝐿 = 95, find m∠𝑆𝐿𝐼., a. 78°, c. 95°, b. 85°, d. 102°, 12. The ________________ angles of a quadrilateral inscribed in a, circle are supplementary., a. adjacent, b. obtuse, c. opposite, d. vertical, 13. All of the following parts from two congruent circles guarantee that two minor, arcs from congruent circles are congruent except for one. Which one is it?, a. Their corresponding congruent chords., b. Their corresponding central angles., c. Their corresponding inscribed angles., d. Their corresponding intercepted arcs., Refer to ⊙O for items 14 and 15., 14. In ⊙O, what is ̅̅̅̅, 𝑃𝑅 if ̅̅̅̅, 𝑁𝑂 = 10 units and ̅̅̅̅, 𝐸𝑆 = 4 units?, a. 64 units, c. 16 units, b. 32 units, d. 8 units, ̂ = 40?, 15. In ⊙O, what is the measure of arc 𝑃𝑅 if 𝑚𝑃𝐸, a. 20°, c. 60°, b. 40°, d. 80°, , 3, , CO_Q2_Mathematics 10_ Module 4

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What’s In, Before we start, let us first have a recap on some parts of a circle., Directions: Rearrange the jumbled letters to come up with a word that corresponds, to the given definition. Write your answers on a separate sheet of paper., 1) C A R – A part of a circle between any two points and is measured in terms of, degrees., 2) R O D C H – A line segment that has its endpoints on the circle., 3) E T E R M A D I – A chord that passes through the center of the circle., 4) L A R T N E C G A N E L – It is angle whose vertex is at the center of a circle, and whose sides are radii of a circle., 5) B C D E I I N R S N E G A L – It is an angle whose vertex lies on the circle and, its sides contain chords of the circle., , Lesson, , 1, , Theorems Related, to Chords, Arcs, and Central, Angles, What’s New, , If and Then, Read the following 𝐼𝑓-𝑡ℎ𝑒𝑛 statements. State whether you agree with the, statement or not. Justify your answer., 1. If an arc measures 180°, then it is a semi-circle., 2. If all radii of a figure are congruent, then the figure is a circle., 3. If an angle is inscribed in a circle, then its measure is 𝑜𝑛𝑒-ℎ𝑎𝑙𝑓 the, measure of its intercepted arc., The activity that you just have done posed situations where a premise is, presented and a conclusion is made. You shall be seeing more of these in lessons 1, and 2 where we will be proving theorems., , 4, , CO_Q2_Mathematics 10_ Module 4

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What I Have Learned, To summarize what you have learned, fill in the blanks with the correct terms., 1. If the radii of the two circles are ___________, then the circles are, congruent., 2. Congruent arcs are arcs of the same circle and of congruent circles with, ___________., 3. Minor arcs of congruent circles having corresponding congruent, ___________ are congruent., , What I Can Do, , On your birthday, your godparent gave you a thousand Pesos as a gift and, told you to spend it wisely. Show, through a budget pie graph, how you would allocate, this amount then answer the questions that follow. Your responses to the questions, and your graph will be scored according to the given rubrics., 1. In which entry was the highest budget allocated? Why did you allot this item, with the highest amount?, 2. In which entry was the least budget allocated? Why did you allot this item, with the least amount?, 3. What is the degree measure of every entry in your pie graph?, 4. How is the measure of the central angles related to the budget you have, allocated for your entries?, Score, 5, 4, 3, 2, 1, , Descriptors for the Content, The justification is correct, substantial, specific, and convincing., The justification is correct, substantial, and specific but not, convincing., The justification is correct and substantial but not specific and, convincing., The justification is correct but not substantial, specific, and, convincing., There is justification but it is not correct, substantial, specific, and, convincing., , 11, , CO_Q2_Mathematics 10_ Module 4

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Score, , 5, 4, 3, 2, 1, , Lesson, , 2, , Descriptors for the Pie Graph, Criteria: a. The budgeting is logical and appropriate., b. The pie graph is accurately divided., c. The pie graph comes with clear and readable descriptions., d. The sum of the amounts in all the entries is ₱1,000., The four criteria were met., Three criteria were met., Two criteria were met., One criterion was met., A pie graph is presented but none of these criteria were met., , Theorems Related, to Arcs, Chords, and, Inscribed Angles, What’s New, , If and Then, Read the following 𝐼𝑓-𝑡ℎ𝑒𝑛 statements. State whether you agree with the, statement or not. Justify your answer., 1. If the circle is intercepted by a diameter, then the arc measures 180⁰., 2. If a square is inscribed in a circle, then it divides the circle into four, congruent arcs., The activity that you just have done posed situations where a premise (the if, clause) is presented and a conclusion (the then clause) is made., , 12, , CO_Q2_Mathematics 10_ Module 4

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What is It, In the next set of activities, you are tasked to prove theorems you used in the, previous module. We are going to review some of them and then provide the proofs to, these theorems., We will start with the following concepts., An inscribed angle is an angle whose vertex is on the circle and whose sides contain, chords of the circle. The arc that lies in the interior of an inscribed angle and has, endpoints on the angle is called the intercepted arc of the angle., An inscribed angle may contain the center of the circle in its interior, may have the, center of the circle on one of its sides, or the center of the circle may be at the exterior, of the circle., Example:, , ∠𝐿𝐴𝑃, ∠𝑇𝑂𝑃, and ∠𝐶𝐺𝑀 are inscribed angles. Their respective vertices, 𝐴, 𝑂, ̅̅̅̅ and, and 𝐺 are points on the circumference of the circles. Their respective sides, 𝐴𝐿, ̅̅̅̅, ̅̅̅̅, ̅̅̅̅, ̅̅̅̅, ̅̅̅̅̅, 𝐴𝑃, 𝑂𝑇 and 𝑂𝑃, and 𝐺𝐶 and 𝐺𝑀, contain chords of the circles., ̂ lie in the interior of inscribed angles ∠𝐿𝐴𝑃, ∠𝑇𝑂𝑃, and ∠𝐶𝐺𝑀,, ̂ , 𝑇𝑃, ̂ , and 𝐶𝑀, 𝐿𝑃, ̂ are the intercepted arcs of these inscribed angles., ̂ , 𝑇𝑃, ̂ , and 𝐶𝑀, respectively. Thus, 𝐿𝑃, Theorems on Inscribed Angles, Theorem 1. If an angle is inscribed in a circle, then the measure of the, angle is equal to 𝑜𝑛𝑒-ℎ𝑎𝑙𝑓 the measure of its intercepted arc., , •, , ̂ is its, In the figure, ∠𝐴𝐶𝑇 is an inscribed angle and 𝐴𝑇, intercepted arc., , •, , ̂ is equal to 120⁰, then the measure of, If the measure of 𝐴𝑇, ∠𝐴𝐶𝑇 is equal to 60⁰., , 13, , CO_Q2_Mathematics 10_ Module 4

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What I Can Do, , In the previous activities, you have done proving using the two-column proof., Based on your daily activities, cite a situation with a justification, where the theorems, on inscribed angles are applied., ___________________________________________________________________________, __________________________________________________________________________________, __________________________________________________________________________________, __________________________________________________________________________________, __________________________________________________________________________________, Score, , Descriptors for each Situation, , 4, , The situation is correct with substantial, specific, and convincing, justification., , 3, , The situation is correct with substantial and specific but not convincing, justification., , 2, , The situation is correct with substantial but not specific and not convincing, justification., , 1, , A situation is presented., , Assessment, Read and analyze each item very carefully. On your answer sheet, write the, letter of the choice that corresponds to the correct answer., 1. Which of the following illustrations do NOT show congruence?, a., c., 𝑟 = 7.25 𝑖𝑛, , 𝑟 = 7.25 𝑖𝑛, , 𝑟 = 7.25 𝑖𝑛, , The two circles are congruent., , 𝑑 = 14.25 𝑖𝑛, , The two circles are congruent., , 18, , CO_Q2_Mathematics 10_ Module 4

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8. What phrase correctly completes the theorem, “If two inscribed angles of a, circle _____, then the angles are congruent”?, a. inscribe congruent arcs, c. inscribed congruent angles, b. intercept congruent arcs, d. intercept congruent angles, ̂ ≅ 𝑃𝐸, ̂ = 30°. Which of the following, ̂ and 𝑀𝑆, 9. In ⊙ 𝑄, 𝑀𝑆, statements is correct?, a. ∠𝑆𝐼𝑀 and ∠𝐸𝐿𝑃 both measure 15°., b. ∠𝑆𝐼𝑀 and ∠𝐸𝐿𝑃 both measure 30°., ̂ and 𝑃𝐸, ̂ inscribe ∠𝑆𝐼𝑀 and ∠𝐸𝐿𝑃., c. 𝑀𝑆, ̂ and 𝑃𝐸, ̂ intercept ∠𝑆𝐼𝑀 and ∠𝐸𝐿𝑃., d. 𝑀𝑆, , 𝑄, , ̂ is a semicircle, 10. In ⊙ 𝐴, what is the measure of ∠𝑆𝐴𝑌 if 𝐷𝑆𝑌, and m∠𝑆𝐴𝐷 = 70?, a. 20°, c. 110°, b. 70°, d. 150°, 11. Quadrilateral 𝑆𝑀𝐼𝐿 is inscribed in ⊙ 𝐸., If m∠𝑆𝑀𝐼 = 78 and m∠𝑀𝑆𝐿 = 95, find m∠𝑀𝐼𝐿., a. 78°, c. 95°, b. 85°, d. 102°, 12. The opposite angles of a quadrilateral inscribed in a circle, are ____., a. complementary b. obtuse, c. right, d. supplementary, 13. All of the following parts from two congruent circles guarantee that two minor, arcs from congruent circles are congruent except for one. Which one is it?, a. Their corresponding congruent chords., b. Their corresponding central angles., c. Their corresponding inscribed angles., d. Their corresponding intercepted arcs., Refer to ⊙O for items 14 and 15., 14. In ⊙O, what is 𝑃𝑅 if 𝑁𝑂 = 15 units and 𝐸𝑆 = 6 units?, a. 28 units, c. 12 units, b. 24 units, d. 9 units, 15. In ⊙O, what is the measure of ∠𝑃𝑆𝑁?, a. 45°, b. 80°, , 20, , c. 90°, d. 180°, , CO_Q2_Mathematics 10_ Module 4

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Additional Activity, , If - then Statement, Compose three original 𝐼𝑓-𝑡ℎ𝑒𝑛 statements. Make sure that your statements, are realistic and acceptable., , 1. __________________________________________________________________, __________________________________________________________________, 2. __________________________________________________________________, __________________________________________________________________, 3. __________________________________________________________________, __________________________________________________________________, Every 𝐼𝑓-𝑡ℎ𝑒𝑛 statement will be scored according to the rubric below., Score, , Descriptors, , 3, , The premise is valid and the conclusion is correct/acceptable., , 2, , The premise is valid but the conclusion is incorrect/unacceptable., , 1, , The premise and the conclusion do not match., , 21, , CO_Q2_Mathematics 10_ Module 4

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CO_Q2_Mathematics 10_ Module 4, What I Know, 1. d, 2. b, 3. c, 4. d, 5. b, 6. c, 7. a, 8. a, , 9. c, 10. a, 11. d, 12. c, 13. d, 14. c, 15. d, , 22, What’s In, 1) ARC, 2) CHORD, 3) DIAMETER, 4) CENTRAL, ANGLE, 5) INSCRIBED, ANGLE, , Lesson 1., What’s New, 1) Yes, definition of, semicircle., 2) Yes because all radii of, the same circle are, congruent., 3) Need to be investigated, , Lesson 1. What’s More, , Activity 1, , Activity 2, , Answer Key

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23, , CO_Q2_Mathematics 10_ Module 4, Lesson 1. What’s More, Activity 3, , The varied outputs, from the students will, be evaluated using the, given rubrics., , 2. equal measures., , Lesson 1. What I, can Do, , Lesson 1. What I, Have Learned, 1. congruent, , 3. central angles, , Lesson 2. What’s New, 1), 2), , Yes because a diameter divides the circle, into two equal parts called semicircles., Yes because a square has four congruent, vertices and these cut the circle into four, congruent arcs., , Lesson 2. What ‘s More, Activity 1.a

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24, , CO_Q2_Mathematics 10_ Module 4, Lesson 2. What’s More, Activity 1.b, , Lesson 2., What’s More, Activity 2, a. 35, b. 55, c. 90, d. 110, e. 180, , Lesson 2., What I can Do, The varied outputs from, the students will be, evaluated, using, the, given rubric., , Lesson 2., What I have Learned, , Activity 3, a. 116, b. 64, c. 116, d. 32, e. 58, , 1. intercepted arc, 2. Opposite angles, , Additional, Activity, , Assessment, 1., 2., 3., 4., 5., , c, c, d, b, a, , 6. d, 7. d, 8. b, 9. a, 10. c, , 11., 12., 13., 14., 15., , c, d, d, b, c, , The varied outputs, from the students, will be evaluated, using the given, rubric.

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References, Nivera, Gladys C., Ph.D. and Minie Rose C. Lapinid, Ph.D.. 2015. Grade 10, Mathematics Patterns and Practicalities. SalesianaBooks. Makati City. Don, Bosco Press, Inc., , Callanta, M.M. Et.Al. Mathematics- Grade 10 Learners Module, 2015, REX, Bookstore, Inc. Pasig City. November 6, 2019., 2015. "Circles." In Mathematics Learner's Module for Grade 10, by Department of, Education, 127 to 177. Pasig City: REX Book Store, Inc., , 25, , CO_Q2_Mathematics 10_ Module 4