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

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

10

Mathematics

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

[illustration: students learning at home, some outdoors under a tree and some at a table with a globe and a lamp]

CO_Q2_Mathematics 10_ Module 4

Mathematics – Grade 10 Alternative Delivery Mode Quarter 2 – Module 4: Proving Theorems Related to Chords, Arcs, Central Angles, and Inscribed Angles First Edition, 2020

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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

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Office Address: Wangal, La Trinidad, Benguet Telefax: (074) 422-4074 E-mail Address: car@deped.gov.ph

10

Mathematics

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

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.

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. [illustration: two circles, each with a horizontal chord labeled r = 8.25 in and r = 8.25 in] The two circles are congruent.

c. [illustration: two circles, one with a chord labeled r = 7.25 in and one with a diameter labeled d = 14.50 in] The two circles are congruent.

b. [illustration: a circle with a 90° central angle at A, with points B and C on the circle and a second circle with a right angle, points C and D labeled] AB̂ ≅ CD̂

d. [illustration: a circle with points E, F, G, H on the circle, with tick marks on chords EF and FH] EF̂ ≅ FĤ

2. An inscribed angle is a right angle if it intercepts a __________.
   a. whole circle b. semicircle c. minor arc d. major arc

What I Know (continued)

3. Consider ⊙ A with EF̂ = 155°. Which of the following statements is always true?
   a. In ⊙ A, ∠EAF = 77.5°.
   b. BĈ = 155° if and only if EF̂ = 155°.
   c. EB̂ ≅ FĈ if and only if EB̅ ≅ FC̅.
   d. ∠EAF ≅ ∠BAF if and only if BCF̂ = 155°.

[illustration: circle with center A, points E, F on the circle, point B and C on the circle, with radii AE and AF and chords BC]

Refer to ⊙ G for items 4 to 6.

4. In ⊙ G, KL̅ is a diameter that is perpendicular to chord HI. Which of the following is true?
   a. KL̅ ≅ HI̅
   b. HK̂ ≅ LÎ
   c. KG̅ ≅ GL̅
   d. HG̅ ≅ GI̅

[illustration: circle with center G, diameter KL vertical, chord HI horizontal, right-angle mark at G]

5. Suppose HI̅ = 15 and IKĤ = 120°, then ____.
   a. HG̅ = 15, GI̅ = 15, KÎ = 120°, and KĤ = 120°
   b. HG̅ = 7.5, GI̅ = 7.5, KÎ = 60°, and KĤ = 60°
   c. KG̅ = 7.5, GL̅ = 7.5, HL̂ = 60°, and LÎ = 60°
   d. Insufficient data, answer cannot be determined.

   
   

6. Suppose HLÎ = 240°, find the measure of KĤ.
   a. 240° b. 120° c. 60° d. 30°

   
   

7. What is the measure of the arc intercepted by inscribed angle ∠NMO if ∠NMO = 85°?
   a. NÔ = 170° c. NMÔ = 42.5°
   b. NMÔ = 170° d. NÔ = 42.5°

   
   

[illustration: circle with points N, M, O on the circle, chords MN and MO]

8. What phrase correctly completes the theorem, “If a quadrilateral is _______________, then its opposite angles are supplementary.”?
   a. inscribed in a circle c. inscribed in a semicircle
   b. circumscribed about a circle d. circumscribed about a semicircle

   
   

9. In ⊙ Q, MŜ ≅ PÊ and MŜ = 40°. Which of the following statements is NOT true?
   a. ∠SIM and ∠ELP both measure 20°.
   b. MŜ and PÊ both measure 40°.
   c. ∠SIM and ∠ELP both measure 40°.
   d. ∠SIM and ∠ELP intercepts arcs MS and EP respectively.

   
   

[illustration: circle with center Q, points S, M, P, E, I, L on the circle, with chords forming inscribed angles]

What I Know (continued)

10. In ⊙ A, what is the measure of ∠SAY if DSŶ is a semicircle and m∠SAD = 50?
    a. 130° c. 100°
    b. 110° d. 50°

[illustration: circle with center A, diameter DY horizontal, point S on the circle, radius AS]

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°

[illustration: circle with center E, quadrilateral with vertices M, I, L, S, angle marks 78° at M and 95° at S]

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 PR̅ if NO̅ = 10 units and ES̅ = 4 units?
    a. 64 units c. 16 units
    b. 32 units d. 8 units

    
    

15. In ⊙O, what is the measure of arc PR if mPÊ = 40?
    a. 20° c. 60°
    b. 40° d. 80°

    
    

[illustration: circle with center O, chord NE passing near S, chords PR and EN, with points P, R, E, S, N labeled]

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.

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. While doing so, note that all images are NOT drawn to scale.

Congruent Circles and Congruent Arcs

Congruent circles are circles with congruent radii.

[illustration: two circles; left circle center Y with radius XY, X on the circle; right circle center A with radius AB, B on the circle]

Example: 𝑋𝑌̅̅̅̅ is a radius of ⊙ 𝑌. 𝐴𝐵̅̅̅̅ is a radius of ⊙ 𝐴. If 𝑋𝑌̅̅̅̅ ≅ 𝐴𝐵̅̅̅̅, then ⊙ 𝑌 ≅ ⊙ 𝐴.

Congruent arcs are arcs of the same circle or of congruent circles with equal measures.

[illustration: circle I with points T (top), M (left), K (bottom), S (right); arcs TM and KS each marked 65º; second circle center X with points Y and Z, arc YZ marked 65º]

Example: In ⊙ 𝐼, if 𝑚𝑇𝑀̂ = 𝑚𝐾𝑆̂, then 𝑇𝑀̂ ≅ 𝐾𝑆̂. If ⊙I ≅ ⊙X and 𝑚𝑇𝑀̂ = 𝑚𝐾𝑆̂ = 𝑚𝑌𝑍̂, then 𝑇𝑀̂ ≅ 𝐾𝑆̂ ≅ 𝑌𝑍̂.

Theorems on Central Angles, Arcs, and Chords

Theorem 1. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.

[illustration: circle P with points N (top), O, M, L; central angles marked 45º (between PN and PO) and 45º (between PL and PM); circle C with points A (top), B; central angle at C marked 45º between CA and CB]

a. In ⊙𝑃, since ∠𝐿𝑃𝑀 ≅ ∠𝑂𝑃𝑁, then 𝑁𝑂̂ ≅ 𝑀𝐿̂. b. If ⊙ 𝑃 ≅ ⊙ 𝐶 and ∠𝐿𝑃𝑀 ≅ ∠𝑂𝑃𝑁 ≅ ∠𝐴𝐶𝐵, then 𝐿𝑀̂ ≅ 𝑂𝑁̂ ≅ 𝐴𝐵̂.

Proof of the Theorem

Use a two-column proof to prove that the intercepted arcs of two corresponding congruent angles from two congruent circles are congruent.

Given: ⊙ 𝑈 ≅ ⊙ 𝐾 and ∠𝑊𝑈𝑆 ≅ ∠𝐿𝐾𝑀 Prove: 𝑊𝑆̂ ≅ 𝐿𝑀̂

[illustration: circle U with points W (left), U (center), S (bottom), angle WUS marked; circle K with points L (upper right), M (right), K (center), angle LKM marked]

Proof:

Statement	Reason
1. ⊙ 𝑈 ≅ ⊙ 𝐾 ∠𝑊𝑈𝑆 ≅ ∠𝐿𝐾𝑀	1. Given
2. In ⊙ 𝑈, m∠𝑊𝑈𝑆 = 𝑚𝑊𝑆̂ In ⊙ 𝐾, m∠𝐿𝐾𝑀 = 𝑚𝐿𝑀̂	2. The measure of a central angle is equal to the degree measure of its intercepted arc.
3. 𝑚∠𝑊𝑈𝑆 = 𝑚∠𝐿𝐾𝑀	3. Congruent angles have equal measures.
4. 𝑚𝑊𝑆̂ = 𝑚𝐿𝑀̂	4. Substitution Property of Equality
5. 𝑾𝑺̂ ≅ 𝑳𝑴̂	5. Two arcs are congruent if they have equal measures.

[transcriber’s note: a stray letter “a” is printed below row 1 in the Statement column, beside the given statements; it is not part of a labeled statement]

Theorem 2 (box at top of page)

Theorem 2. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

[illustration: circle T with points A (top), B (left), H (right), C (bottom), chords AB and HC drawn; circle N with points O and E at bottom, chord OE drawn]

a. In ⊙ 𝑇, 𝐵𝐴̅̅̅̅ ≅ 𝐶𝐻̅̅̅̅. Since the two chords are congruent, then 𝐵𝐴̂ ≅ 𝐶𝐻̂. b. If ⊙ 𝑇 ≅ ⊙ 𝑁 and 𝐵𝐴̅̅̅̅ ≅ 𝐶𝐻̅̅̅̅ ≅ 𝑂𝐸̅̅̅̅, then 𝐵𝐴̂ ≅ 𝐶𝐻̂ ≅ 𝑂𝐸̂.

Proof of the Theorem

Given that ⊙ 𝑇 ≅ ⊙ 𝑁 and 𝐴𝐵̅̅̅̅ ≅ 𝑂𝐸̅̅̅̅, use a two-column proof to prove that 𝐴𝐵̂ and 𝑂𝐸̂ are congruent.

Given: ⊙ 𝑇 ≅ ⊙ 𝑁 𝐴𝐵̅̅̅̅ ≅ 𝑂𝐸̅̅̅̅ Prove: 𝐴𝐵̂ ≅ 𝑂𝐸̂

[illustration: circle T with points A (top), B (left), H (right), C (bottom), center T; circle N with points O and E on the bottom chord]

Proof:

Statement	Reason
1. ⊙ T ≅ ⊙ N 𝐴𝐵̅̅̅̅ ≅ 𝑂𝐸̅̅̅̅	1. Given
2. 𝑇𝐴̅̅̅̅ ≅ 𝑇𝐵̅̅̅̅ ≅ 𝑁𝑂̅̅̅̅ ≅ 𝑁𝐸̅̅̅̅	2. Radii of the same circle or of congruent circles are congruent.
3. ∆𝐴𝑇𝐵 ≅ ∆𝑂𝑁𝐸	3. SSS Postulate
4. ∠𝐴𝑇𝐵 ≅ ∠𝑂𝑁𝐸	4. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
5. 𝑨𝑩̂ ≅ 𝑶𝑬̂	5. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.

Theorem 3 (box at top of page)

Theorem 3. In a circle, a diameter bisects a chord and an arc with the same endpoints if and only if it is perpendicular to the chord.

[chart: circle diagram; center M, diameter QR horizontal, chord ST vertical and perpendicular to QR at point where QR meets ST; points Q (left), R (right), S (top), T (bottom); each half of the chord labeled 3 𝑐𝑚; arcs SR and RT each labeled 40°. Values are as printed.]

In ⊙ 𝑀, diameter 𝑄𝑅 bisects chord 𝑆𝑇 and 𝑆𝑇̂ since 𝑄𝑅̅̅̅̅ ⊥ 𝑆𝑇̅̅̅̅.

Proof of the Theorem

Use a two-column proof to prove that segments and arcs are congruent by showing that 𝐴𝑍̅̅̅̅ bisects 𝐽𝑀̅̅̅̅ and 𝐽𝑀̂.

Given: In ⊙W, 𝐴𝑍̅̅̅̅ is a diameter. 𝐴𝑍̅̅̅̅ ⊥ 𝐽𝑀̅̅̅̅ at T. Prove: 1. 𝐴𝑍̅̅̅̅ bisects 𝐽𝑀̅̅̅̅ 2. 𝐴𝑍̅̅̅̅ bisects 𝐽𝑀̂

[illustration: circle W with points J (top), A (left), T (intersection point), W (center), Z (right), M (bottom); diameter AZ horizontal and perpendicular to chord JM at T]

Proof:

Statements	Reasons
1. ⊙W with diameter 𝐴𝑍̅̅̅̅ ⊥ chord 𝐽𝑀̅̅̅̅	1. Given
2. ∠𝐽𝑇𝑊 and ∠𝑀𝑇𝑊 are right angles.	2. Definition of Perpendicular Lines
3. ∠𝐽𝑇𝑊 ≅ ∠𝑀𝑇𝑊	3. Right angles are congruent.
4. 𝑊𝐽̅̅̅̅ ≅ 𝑊𝑀̅̅̅̅̅	4. Radii of the same circle are congruent.
5. 𝑊𝑇̅̅̅̅̅ ≅ 𝑊𝑇̅̅̅̅̅	5. Reflexive/Identity Property of Equality
6. ∆𝐽𝑇𝑊 ≅ ∆𝑀𝑇𝑊	6. HyL Theorem
7. 𝐽𝑇̅̅̅ ≅ 𝑀𝑇̅̅̅̅̅	7. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
8. 𝑨𝒁̅̅̅̅ bisects 𝑱𝑴̅̅̅̅	8. Definition of Segment Bisector
9. ∠𝐽𝑊𝐴 ≅ 𝑚∠𝑀𝑊𝐴	9. CPCTC
10. 𝑚∠𝐽𝑊𝐴 = 𝑚∠𝑀𝑊𝐴	10. Congruent angles have equal measures.
11. 𝑚𝐴𝐽̂ = 𝑚∠𝐽𝑊𝐴 𝑚𝐴𝑀̂ = 𝑚∠𝑀𝑊𝐴	11. The degree measure of an arc and the central angle that intercepts it are equal.
12. 𝑚𝐴𝑀̂ = 𝑚𝐴𝐽̂	12. Substitution Property of Equality
13. 𝐴𝑀̂ ≅ 𝐴𝐽̂	13. Definition of Congruent Arcs
14. 𝑨𝒁̅̅̅̅ bisects 𝑱𝑴̂	Definition of Segment Bisector

What’s More

Activity 1. Prove Me Right

Complete the two-column proof to prove that the central angles intercepting two corresponding congruent minor arcs from corresponding congruent circles are congruent. Be guided by the statements and reasons already provided for you.

Given: ⊙ 𝑈 ≅ ⊙ 𝐾 and 𝑊𝑆̂ ≅ 𝐿𝑀̂ Prove: ∠𝑊𝑈𝑆 ≅ ∠𝐿𝐾𝑀

[illustration: circle U with points W (left), U (center), S (bottom), angle WUS marked; circle K with points L (upper right), M (right), K (center), angle LKM marked]

Proof:

Statement	Reason
1.	1. Given
2. In ⊙ 𝑈, 𝑚𝑊𝑆̂ = 𝑚∠𝑊𝑈𝑆. In ⊙ 𝐾, 𝑚𝐿𝑀̂ = 𝑚∠𝐿𝐾𝑀.	2.
3.	3. Definition of Congruent Arcs
4. 𝑚∠𝑊𝑈𝑆 = 𝑚∠𝐿𝐾𝑀	4.
5. ∠𝑾𝑼𝑺 ≅ ∠𝑳𝑲𝑴	5.

Activity 2. Minor Arcs and Chords

Complete the two-column proof to prove that the chords from congruent circles with corresponding congruent minor arcs are congruent. Be guided by the statements and reasons already provided for you.

Given: ⊙ 𝑇 ≅ ⊙ 𝑁 𝐴𝐵̂ ≅ 𝑂𝐸̂ Prove: 𝐴𝐵̅̅̅̅ ≅ 𝑂𝐸̅̅̅̅

[illustration: circle T with points A (top), B (left), H (right), C (bottom), center T; circle N with points O and E on the bottom chord]

Proof:

Statement	Reason
1.	1. Given
2. 𝑚𝐴𝐵̂ = 𝑚𝑂𝐸̂	2.
3. 𝑚𝐴𝐵̂ = 𝑚∠𝐴𝑇𝐵 and 𝑚𝑂𝐸̂ = 𝑚∠𝑂𝑁𝐸	3.
4.	4. Substitution Property of Equality
5. ∠𝐴𝑇𝐵 ≅ ∠𝑂𝑁𝐸	5.
6. 𝑇𝐴̅̅̅̅ ≅ 𝑇𝐵̅̅̅̅ ≅ 𝑁𝑂̅̅̅̅ ≅ 𝑁𝐸̅̅̅̅	6.
7.	7. SAS Postulate
8. 𝑨𝑩̅̅̅̅ ≅ 𝑶𝑬̅̅̅̅	8.

Activity 3. Diameter Bisects Chords.

Complete the two-column proof to prove that the two chords are perpendicular if the diameter bisects the other chord. Be guided by the statements and reasons already provided for you.

Given: ⊙𝑈 with diameter 𝐸𝑆̅̅̅̅ 𝐸𝑆̅̅̅̅ bisects 𝐺𝑁̅̅̅̅ at I 𝐸𝑆̅̅̅̅ bisects 𝐺𝑁̂ at 𝐸 Prove: 𝐸𝑆̅̅̅̅ ⊥ 𝐺𝑁̅̅̅̅

[illustration: circle with center U; diameter ES horizontal with E (left), U (center), S (right); chord GN vertical with G (top), I (on ES, where chord meets diameter), N (bottom)]

Proof:

Statement	Reasons
1.	1. Given
2. 𝐺𝐼̅̅̅ ≅ 𝑁𝐼̅̅̅̅ 𝐺𝐸̂ = 𝑁𝐸̂	2.
3. 𝑈𝐼̅̅̅ ≅ 𝑈𝐼̅̅̅	3.
4.	4. Radii of the same circle are congruent.
5. ∆𝐺𝐼𝑈 ≅ ∆𝑁𝐼𝑈	5.
6. ∠𝑈𝐼𝐺 ≅ ∠𝑈𝐼𝑁	6.
7. ∠𝑈𝐼𝐺 and ∠𝑈𝐼𝑁 are right angles.	7. Angles which form a linear pair and are congruent are right angles.
8. 𝐼𝑈̅̅̅ ⊥ 𝐺𝑁̅̅̅̅	8.
9. 𝐸𝑆̅̅̅̅ ⊥ 𝐺𝑁̅̅̅̅	9. 𝐼𝑈̅̅̅ is on 𝐸𝑆̅̅̅̅

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	Descriptors for the Content
5	The justification is correct, substantial, specific, and convincing.
4	The justification is correct, substantial, and specific but not convincing.
3	The justification is correct and substantial but not specific and convincing.
2	The justification is correct but not substantial, specific, and convincing.
1	There is justification but it is not correct, substantial, specific, and convincing.

Score	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.
5	The four criteria were met.
4	Three criteria were met.
3	Two criteria were met.
2	One criterion was met.
1	A pie graph is presented but none of these criteria were met.

Lesson 2 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.

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:

[illustration: Figure 1, circle with points L, A, P, chords AL and AP; Figure 2, circle with points T, O, P, chords OT and OP; Figure 3, circle with points G, C, M, chords GC and GM]

Figure 1 · Figure 2 · Figure 3

∠𝐿𝐴𝑃, ∠𝑇𝑂𝑃, and ∠𝐶𝐺𝑀 are inscribed angles. Their respective vertices, 𝐴, 𝑂 and 𝐺 are points on the circumference of the circles. Their respective sides, 𝐴𝐿̅̅̅̅ and 𝐴𝑃̅̅̅̅, 𝑂𝑇̅̅̅̅ and 𝑂𝑃̅̅̅̅, and 𝐺𝐶̅̅̅̅ and 𝐺𝑀̅̅̅̅̅, contain chords of the circles.

𝐿𝑃̂, 𝑇𝑃̂, and 𝐶𝑀̂ lie in the interior of inscribed angles ∠𝐿𝐴𝑃, ∠𝑇𝑂𝑃, and ∠𝐶𝐺𝑀, respectively. Thus, 𝐿𝑃̂, 𝑇𝑃̂, and 𝐶𝑀̂ are the intercepted arcs of these inscribed angles.

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.

• In the figure, ∠𝐴𝐶𝑇 is an inscribed angle and 𝐴𝑇̂ is its intercepted arc.

  
  

  [illustration: circle with points A, C, T; chords CA and CT, with center marked]

  
  

• If the measure of 𝐴𝑇̂ is equal to 120⁰, then the measure of ∠𝐴𝐶𝑇 is equal to 60⁰.

  
  

13 CO_Q2_Mathematics 10_ Module 4 (printed page number omitted)

Proof of the Theorem

Given: ∠𝑃𝑄𝑅 is inscribed in ⊙𝑆 and 𝑃𝑄̅̅̅̅ is a diameter.

Prove: 𝑚∠𝑃𝑄𝑅 = 1/2 𝑚𝑃𝑅̂

Proof:

Draw 𝑃𝑄̅̅̅̅ and let 𝑚∠𝑃𝑄𝑅 = 𝑥

[illustration: circle S with points P, Q, R, and center S; chords PQ, QR, PR, with segment SR dashed]

Statement	Reasons
1. ∠𝑃𝑄𝑅 is inscribed in ⊙𝑆 and 𝑃𝑄̅̅̅̅ is a diameter.	1. Given
2. 𝑄𝑆̅̅̅̅ ≅ 𝑅𝑆̅̅̅̅	2. Radii of a circle are congruent.
3. ∆QRS is an isosceles ∆.	3. Definition of Isosceles Triangle
4. ∠𝑃𝑄𝑅 ≅ ∠𝑄𝑅𝑆	4. The base angles of an isosceles triangle are congruent.
5. 𝑚∠𝑃𝑄𝑅 = 𝑚∠𝑄𝑅𝑆	5. The measures of congruent angles are equal.
6. 𝑚∠𝑄𝑅𝑆 = 𝑥	6. Transitive Property of Equality (If 𝑚∠𝑃𝑄𝑅 = 𝑥 and 𝑚∠𝑃𝑄𝑅 = 𝑚∠𝑄𝑅𝑆, then 𝑚∠𝑄𝑅𝑆 = 𝑥.
7. 𝑚∠𝑃𝑆𝑅 = 2𝑥	7. The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.
8. 𝑚∠𝑃𝑆𝑅 = 𝑚𝑃𝑅̂ a	8. The measure of a central angle is equal to the measure of its intercepted arc. a
9. 𝑚𝑃𝑅̂ = 2𝑥 a	9. Transitive Property of Equality (from 7 & 8) a
10. 𝑚𝑃𝑅̂ = 2(𝑚∠𝑃𝑄𝑅) a	10. Substitution Property of Equality (from 5 & 6) a
11. 𝒎∠𝑷𝑸𝑹 = 𝟏/𝟐 𝒎𝑷𝑹̂ a	11. Multiplication Property of Equality

[transcriber’s note: a lone “a” is printed at the end of rows 8 to 11 of the proof table; it is transcribed as it appears]

Theorem 2. If two inscribed angles of a circle (or of congruent circles) intercept congruent arcs or the same arc, then the angles are congruent.

In figure 1, ∠𝑃𝐼𝑂 and ∠𝑃𝐿𝑂 intercept 𝑃𝑂̂. Since ∠𝑃𝐼𝑂 and ∠𝑃𝐿𝑂 intercept the same arc, then ∠𝑃𝐼𝑂 ≅ ∠𝑃𝐿𝑂.

[illustration: Figure 1, circle with center T and points P, I, L, O; chords joining them]

Figure 1

In figure 2, ∠𝑆𝐼𝑀 and ∠𝐸𝐿𝑃 intercept 𝑆𝑀̂ and 𝐸𝑃̂, respectively. If 𝑆𝑀̂ is congruent to 𝐸𝑃̂, then ∠𝑆𝐼𝑀 ≅ ∠𝐸𝐿𝑃.

[illustration: Figure 2, circle with points S, I, L, E, M, P and center marked]

Figure 2

The proof of the theorem is given as an exercise in activity 1 of what’s more.

Example 1. ∆GOA is inscribed in ⊙𝐿. If the measurement of ∠𝑂𝐺𝐴 = 75 and the measure of 𝐴𝐺̂ is 160°, find:

a. 𝑚𝑂𝐴̂

𝑚∠𝑂𝐺𝐴 = 1/2 𝑚𝑂𝐴̂

75 = 1/2 𝑚𝑂𝐴̂

150 = 𝑚𝑂𝐴̂

𝑚𝑂𝐴̂ = 150

b. m∠𝐺𝑂𝐴

𝑚∠𝐺𝑂𝐴 = 1/2 𝑚𝐴𝐺̂

𝑚∠𝐺𝑂𝐴 = 1/2 (160)

𝑚∠𝐺𝑂𝐴 = 80

[illustration: circle L with triangle GOA inscribed; angle 75° at G, arc AG labeled 160°]

Theorem 3. If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle.

In ⊙𝑂, ∠𝑁𝑇𝐸 intercepts 𝑁𝑆𝐸̂. If 𝑁𝑆𝐸̂ is a semicircle, then ∠𝑁𝑇𝐸 is a right angle.

[illustration: circle O with points N, S, T, E; chords NT and TE, with center O]

Proof of the Theorem

Given: in circle O, ∠𝑁𝑇𝐸 intercepts a semicircle 𝑁𝑆𝐸

Prove: ∠𝑁𝑇𝐸 is a right angle

Proof:

Statements	Reasons
1. ∠𝑁𝑇𝐸 intercepts a semicircle 𝑁𝑆𝐸	Given
2. 𝑚𝑁𝑆𝐸̂ = 180°	The degree measure of a semicircle is 180°.
3. ∠𝑁𝑇𝐸 = 90°	The degree measure of an inscribed angle is one-half the degree measure of its intercepted arc.
4. ∠𝑁𝑇𝐸 is a right angle	If angle measures 90°, then it is a right angle.

If Quadrilateral 𝐷𝑅𝐸𝐴 is inscribed in ⊙𝑀, then

• 𝑚∠𝑅𝐷𝐴 + 𝑚∠𝑅𝐸𝐴 = 180.
• 𝑚∠𝐷𝑅𝐸 + 𝑚∠𝐷𝐴𝐸 = 180.

[illustration: circle M with quadrilateral DREA inscribed; points R, D, A, E]

The proof of the theorem is given as an exercise in activity 1 of what’s more.

Example 2. Quadrilateral 𝐹𝐴𝐼𝑇 is inscribed in ⊙𝐻. If 𝑚∠𝐴𝐹𝑇 = 75 and 𝑚∠𝐹𝑇𝐼 = 98, find:

a. 𝑚∠𝑇𝐼𝐴

180⁰ = 𝑚∠𝐴𝐹𝑇 + 𝑚∠𝑇𝐼𝐴

180⁰ = 75⁰ + 𝑚∠𝑇𝐼𝐴

180⁰ − 75⁰ = 𝑚∠𝑇𝐼𝐴

105⁰ = 𝑚∠𝑇𝐼𝐴

𝑚∠𝑇𝐼𝐴 = 105

b. 𝑚∠𝐹𝐴𝐼

180⁰ = 𝑚∠𝐹𝑇𝐼 + 𝑚∠𝐹𝐴𝐼

180⁰ = 98⁰ + 𝑚∠𝐹𝐴𝐼

180⁰ − 98⁰ = 𝑚∠𝐹𝐴𝐼

82⁰ = 𝑚∠𝐹𝐴𝐼

𝑚∠𝐹𝐴𝐼 = 82

[illustration: circle H with quadrilateral FAIT inscribed; 75° at F, 98° at T]

Theorem 4. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

What’s More

Activity 1. Prove me right. Write a proof for each of the following theorems.

a) If two inscribed angles of a circle (or of congruent circles) intercept congruent arcs or the same arc, then the angles are congruent.

Given: In the circle at the right, 𝑆𝑀̂ and 𝑃𝐸̂ are the intercepted arcs of ∠𝑆𝐼𝑀 and ∠𝐸𝐿𝑃 respectively. 𝑆𝑀̂ ≅ 𝑃𝐸̂

Prove: ∠𝑆𝐼𝑀 ≅ ∠𝐸𝐿𝑃

[illustration: circle with points S, I, L, E, M, P; chords forming ∠SIM and ∠ELP]

b) If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

Given: Quadrilateral DREA is inscribed in ⊙ 𝑀.

Prove: ∠𝑅𝐷𝐴 and ∠𝑅𝐸𝐴 are supplementary.

[illustration: circle M with quadrilateral DREA inscribed; points R, D, A, E]

Activity 2. Read and Analyze

𝐷𝑅̅̅̅̅ is a diameter of ⊙𝑂. If 𝑚𝑀𝑅̂ = 70, find:

a. 𝑚∠𝑅𝐷𝑀 b. 𝑚∠𝐷𝑅𝑀 c. 𝑚∠𝐷𝑀𝑅 d. 𝑚𝐷𝑀̂ e. 𝑚𝑅𝐷̂

[illustration: circle O with diameter DR, points M and D, arc MR labeled 70°]

Activity 3. A Quad!

Rectangle 𝑇𝐸𝐴𝑀 is inscribed in ⊙𝐵. If 𝑚𝑇𝐸̂ = 64 and 𝑚∠𝑇𝐸𝑀 = 58, find:

a. 𝑚𝑇𝑀̂ b. 𝑚𝑀𝐴̂ c. 𝑚𝐴𝐸̂ d. 𝑚∠𝑀𝐸𝐴 e. 𝑚∠𝑇𝐴𝑀

[illustration: circle B with rectangle TEAM inscribed and diagonals drawn]

How did you do in the activity? What did you find out? I believe you learned something and discovered or proved that you are able to provide correct conclusions backed up by valid reasons.

What I Have Learned

After doing the activities, summarize what you have learned by filling in the blanks with the correct terms.

1. The measure of an inscribed angle is one-half the measure of its _____________.
2. _____________ of an inscribed quadrilateral are supplementary.

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. [illustration: two congruent circles, each labeled r = 7.25 in] The two circles are congruent.

c. [illustration: two circles, one labeled r = 7.25 in and the other labeled d = 14.25 in] The two circles are congruent.

b. [illustration: two circles, central angle of 90° at A with arc AB and chord CD marked with a right angle] 𝐴𝐵̂ ≅ 𝐶𝐷̂

d. [illustration: circle with chords EF and GH marked congruent by tick marks] 𝐸𝐹̂ ≅ 𝐺𝐻̂

2. If an inscribed angle of a circle intercepts a semicircle, the angle is ____.
   a. acute b. obtuse c. right d. straight

   
   

3. Consider ⊙ 𝐴 with 𝐸𝐹̂ = 125°. Which of the following statements is NOT always true?

   
   

   a. In ⊙ 𝐴, ∠𝐸𝐴𝐹 = 125°.
   b. 𝐵𝐶̂ = 125° if and only if 𝐸𝐹̅̅̅̅ ≅ 𝐵𝐶̅̅̅̅.
   c. 𝐸𝐹̂ ≅ 𝐵𝐹̂ if and only if ∠𝐸𝐴𝐹 ≅ ∠𝐵𝐴𝐹.
   d. ∠𝐸𝐴𝐹 ≅ ∠𝐵𝐴𝐹 if and only if 𝐸𝐹̂ = 125°.

   
   

   [illustration: circle A with points E, F, B, C; central angle EAF]

   
   

Refer to ⊙ 𝐺 for items 4 to 6.

4. Which of the following best describes the illustration involving ⊙ 𝐺?
   a. 𝐾𝐿̅̅̅̅ is bisected at 𝐺.
   b. 𝐻𝐼̅̅̅̅ is bisected at 𝐺.
   c. Any two intersecting diameters are perpendicular.
   d. When diameters are perpendicular, they intersect at the center of the circle.

   
   

   [illustration: circle G with perpendicular diameters HI and KL meeting at G, right angle marked]

   
   

5. Suppose 𝐻𝐼̅̅̅̅ = 14.5 and 𝐼𝐾𝐻̂ = 105°, then ____.
   a. 𝐻𝐺̅̅̅̅ = 7.25, 𝐺𝐼̅̅̅ = 7.25, 𝐾𝐼̂ = 52.5°, and 𝐾𝐻̂ = 52.5°
   b. 𝐻𝐺̅̅̅̅ = 14.5, 𝐺𝐼̅̅̅ = 14.5, 𝐾𝐼̂ = 105°, and 𝐾𝐻̂ = 105°
   c. 𝐾𝐺̅̅̅̅ + 𝐺𝐿̅̅̅̅ = 29 and 𝐻𝐼̂ + 𝐿𝐼̂ = 255°.
   d. Insufficient data, answer cannot be determined.

   
   

6. Suppose 𝐻𝐿𝐼̂ = 200°, find the measure of 𝐾𝐻̂.
   a. 360° b. 200° c. 160° d. 80°

   
   

7. What is the measure of the arc intercepted by inscribed angle ∠𝑁𝑀𝑂 if ∠𝑁𝑀𝑂 = 60°?
   a. 𝑁𝑂̂ = 30° c. 𝑁𝑀𝑂̂ = 120°
   b. 𝑁𝑀𝑂̂ = 30° d. 𝑁𝑂̂ = 120°

   
   

   [illustration: circle with inscribed angle at M formed by chords MN and MO]

   
   

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

   
   

9. In ⊙ 𝑄, 𝑀𝑆̂ ≅ 𝑃𝐸̂ and 𝑀𝑆̂ = 30°. Which of the following statements is correct?
   a. ∠𝑆𝐼𝑀 and ∠𝐸𝐿𝑃 both measure 15°.
   b. ∠𝑆𝐼𝑀 and ∠𝐸𝐿𝑃 both measure 30°.
   c. 𝑀𝑆̂ and 𝑃𝐸̂ inscribe ∠𝑆𝐼𝑀 and ∠𝐸𝐿𝑃.
   d. 𝑀𝑆̂ and 𝑃𝐸̂ intercept ∠𝑆𝐼𝑀 and ∠𝐸𝐿𝑃.

   
   

   [illustration: circle Q with points S, I, L, E, M, P; chords forming ∠SIM and ∠ELP]

   
   

10. In ⊙ 𝐴, what is the measure of ∠𝑆𝐴𝑌 if 𝐷𝑆𝑌̂ is a semicircle and m∠𝑆𝐴𝐷 = 70?
    a. 20° c. 110°
    b. 70° d. 150°

    
    

    [illustration: circle A with diameter DY, points S and D, center A]

    
    

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°

    
    

    [illustration: circle E with quadrilateral SMIL inscribed; 78° at M, 95° at S]

    
    

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° c. 90°
    b. 80° d. 180°

    
    

[illustration: circle O with chord PR, perpendicular diameter EN meeting PR at S, right angle marked]

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.

Answer Key

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

    
    

Lesson 1.

What’s In

1. ARC

   
   

2. CHORD

   
   

3. DIAMETER

   
   

4. CENTRAL ANGLE

   
   

5. INSCRIBED ANGLE

   
   

What’s New

1. Yes, definition of semicircle.

   
   

2. Yes because all radii of the same circle are congruent.

   
   

3. Need to be investigated.

   
   

What’s More Lesson 1

Activity 1

Statement	Reason
1. ⊙U ≅ ⊙K; ⌒WS ≅ ⌒LM	1. Given
2. In ⊙U, m⌒WS = m∠WUS. In ⊙K, m⌒LM = m∠LKM.	2. The degree measure of a minor arc is equal to the measure of the central angle which intercepts the arc.
3. m⌒WS = m⌒LM	3. Definition of Congruent Arcs
4. m∠WUS = m∠LKM	4. Substitution Property of Equality
5. ∠WUS ≅ ∠LKM	5. Definition of Congruent Angles

Activity 2

Statement	Reason
1. ⊙T ≅ ⊙N; AB ≅ OE	1. Given
2. m⌒AB ≅ m⌒OE	2. Definition of Congruent Arcs
3. m⌒AB = m∠ATB and m⌒OE = m∠ONE	3. The degree measure of a minor arc is equal to the measure of the central angle which intercepts it.
4. m∠ATB = m∠ONE	4. Substitution Property of Equality
5. ∠ATB ≅ ∠ONE	5. Definition of Congruent Angles
6. TA ≅ TB ≅ NO ≅ NE	6. The radii of the same circle or of congruent circles are congruent.
7. ΔATB ≅ ΔONE	7. SAS Postulate
8. AB ≅ OE	8. CPCTC

Lesson 1.

What’s More

Activity 3

Statement	Reasons
1. ⊙U with diameter ES, ES bisects GN at I [?], and ES bisects GN	1. Given
2. GI ≅ NI; GE = NE [?]	2. Definition of Bisector
3. UI ≅ UI	3. Reflexive/Identity Property of Equality
4. UG ≅ UN	4. Radii of the same circle are congruent.
5. ΔGIU ≅ ΔNIU	5. SSS Postulate
6. ∠GIU = ∠NIU	6. CPCTC
7. ∠GIU and ∠UIN are right angles.	7. Angles which form linear pair and are congruent are right angles.
8. UI ⊥ GN	8. Definition of Perpendicular Lines
9. IU is on ES	[illegible]
9. IU ≅ ES [?]	9. IU is on ES [?]

What I Have Learned

1. congruent

   
   

2. equal measures.

   
   

3. central angles

   
   

What I can Do

The varied outputs from the students will be evaluated using the given rubrics.

Lesson 2.

What’s New

1. Yes because a diameter divides the circle into two equal parts called semicircles.

   
   

2. Yes because a square has four congruent vertices and these cut the circle into four congruent arcs.

   
   

What’s More

Activity 1.a

Statement	Reason
1. In the circle, SM and PE are the intercepted arcs of ∠SIM and ∠ELP respectively and SM ≅ PE	Given
2. m⌒SIM = m⌒PE	2. Definition of Congruent Arcs
3. m∠SIM = ½ m⌒SM; m∠ELP = ½ m⌒PE	3. The degree measure of an inscribed angle is one-half the degree measure of its intercepted arc.
4. 2m∠SIM = m⌒SM; 2m∠ELP = m⌒PE	4. Multiplication property of equality
5. 2m∠SIM = 2m∠ELP	5. Substitution property of equality
6. m∠SIM = m∠ELP	6. Division property of equality
7. ∠SIM ≅ ∠ELP	7. Congruent angles are angles with equal measures.

What’s More

Activity 1.b

Statement	Reasons
1. Quadrilateral DREA is inscribed in ⊙M	1. Given
2. m⌒REA + m⌒RDA = 360	2. REA and RDA formed a circle.
3. ∠RDA intercepts REA and ∠REA intercepts RDA	3. Definition of intercepted arc
4. m∠RDA = ½ m⌒REA and m∠REA = ½ m⌒RDA	4. The degree measure of an inscribed angle is one-half the degree measure of its intercepted arc.
5. 2m∠RDA = m⌒REA and 2m∠REA = m⌒RDA	5. Multiplication property of equality
6. 2m∠RDA + 2m∠REA = 360	6. Substitution Property of Equality
7. m∠RDA + m∠REA = 180	7. Division Property of Equality
8. ∠RDA and ∠REA are supplementary	8. Definition of supplementary angles

Activity 2

1. 35

   
   

2. 55

   
   

3. 90

   
   

4. 110

   
   

5. 180

   
   

Activity 3

1. 116

   
   

2. 64

   
   

3. 116

   
   

4. 32

   
   

5. 58

   
   

What I Have Learned

1. intercepted arc

   
   

2. Opposite angles

   
   

What I can Do

The varied outputs from the students will be evaluated using the given rubric.

Assessment

1. c
2. c
3. d
4. b
5. a
6. d
7. d
8. b
9. a
10. c
11. c
12. d
13. d
14. b
15. c

Additional Activity

The varied outputs from the students will be evaluated using the given rubric.

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.

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