Republic of the Philippines

Department of Education Regional Office IX, Zamboanga Peninsula

Zest for Progress

Zeal of Partnership

8

4th QUARTER – Module 2: APPLICATIONS OF TRIANGLE

INEQUALITIES

Name of Learner: ___________________________

Grade & Section: ___________________________

Name of School: ___________________________

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Mathematics – Grade 8 Alternative Delivery Mode Quarter 4 - Module 2: Applications of Triangle Inequality First Edition, 2020

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Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Printed in the Philippines

Department of Education – Region IX, Zamboanga Peninsula

Office Address: Tiguma, Airport Road, Pagadian City

Telefax: (062) – 215 – 3751; 991 – 5975

E-mail Address: region9@deped.gov.ph

Development Team of the Module

Writer: Gizelle Narcitta S. Jasim

Editors: Eleanor G. Cayona

Viola I. Quiniquito

Layout Artist: Abdurauf J. Baldomero

Reviewers: EPS, Mathematics Vilma A. Brown, Ed. D.

Principal Garry D. Sta. Ana

Management Team: SDS Roy C. Tuballa, EMD, JD, CESO VI

ASDS Jay S. Montealto, CESO VI

ASDS Norma T. Francisco, DM, CESE

EPS Mathematics Vilma A. Brown, Ed. D.

EPS LRMS Aida F. Coyme, Ed. D.

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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-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 can 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. 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.

The main goal of this module is to help students to practice their skills in the different

theorems on triangle inequalities.

At the end of this module, you are expected to apply theorems on triangle inequalities.

(M8GE-IVb-1)

What I Know

Directions: Read each item carefully. Write the letter of the correct answer on a separate

sheet of paper.

1. Anna, Joan, Mary, and Amy were given a 20-inch piece of stick. They were instructed to

create a triangle.They cut the stick in their own chosen lengths as follows: Anna—5 in, 9

in, 6 in; Joan—2 in, 2 in, 16 in; Mary—7 in, 7 in, 6 in; and Amy—9 in, 8 in, 3 in. Who among

them was not able to make a triangle?

A. Anna B. Joan C. Mary D. Amy

2. Two sides of a triangle have lengths 7 and 12. Which inequalities represent the possible

lengths for the third side, x?

A. 5 < 𝑥 < 12 B. 5 > 𝑥 > 12 C. 5 > 𝑥 > 19 D. 5 < 𝑥 < 19

What I Need to Know

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3. Which three lengths CANNOT be the lengths of the sides of a triangle?

A. 23 m, 17 m, 14 m C. 11 m, 11 m, 12 m

B. 5 m, 7 m, 8 m D. 21 m, 4 m, 4 m

4. What is the measurement of the exterior angle of an equilateral triangle?

A. 120o B. 100o C. 60o D. 30o

5. Which of the following does not represent the lengths of the sides of a triangle?

A. 5cm,2cm,1cm B. 6cm,3cm,9cm C. 5cm,7cm,8cm D. 3cm,3cm,3cm

What’s In

Direction: Match the illustrations in column B with the theorems in column A. Write the letter of the correct answer on a separate sheet.

THEOREM ILLUSTRATION 1. Exterior Angle Inequality

Theorems The measure of an

exterior angle of a triangle is greater than the measure of any of its remote interior angles.

2. Triangle Inequality Theorem 1 (Ss→Aa)

If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side.

3. Triangle Inequality Theorem 2 (Aa→Ss)

If one angle of a triangle is larger than a second angle, then the side opposite the first angle is longer than the side opposite the second angle.

4. Lesson 4. Triangle Inequality Theorem 3 (S1 + S2 > S3)

5. Hinge Theorem (SAS Triangle Inequality Theorem)

If two triangles have two congruent sides, then the one with the larger included angle has the longer third side.

A. Suppose in ∆𝐴𝐵𝐶 and

∆𝑋𝑌𝑍 , 𝐴𝐵̅̅ ̅̅ ≅ 𝑋𝑌̅̅ ̅̅ and

𝐵𝐶̅̅ ̅̅ ≅ 𝑌𝑍̅̅̅̅ . We can say that

𝐴𝐶̅̅ ̅̅ is greater than 𝑌𝑍̅̅̅̅ since ∆𝐴𝐵𝐶 has a larger included angle than ∆𝑋𝑌𝑍.

B. Note that in a triangle,

• The smallest angle is opposite the shortest side.

• The larger angle is opposite the longer side.

• The largest angle is opposite the longest side

C. Let 𝒂 = 𝟑, 𝒃 = 𝟒, and 𝒄 = 𝟏𝟎.

𝑎 + 𝑏 > 𝑐 3 + 4 > 10 or 7 > 10 (False) 𝑎 + 𝑐 > 𝑏 3 + 10 > 4 or 13 > 4 (True)

𝑏 + 𝑐 > 𝑎 4 + 10 > 3 or 14 > 3 (True) Since one of the situations is false, therefore, we can

conclude that we cannot form a triangle with side lengths 3, 4, and 10 units.

D.

∠ACD=1000

E. In ∆𝑋𝑌𝑍 , if 𝑋𝑍̅̅ ̅̅ > 𝑌𝑍̅̅̅̅ and

𝑌𝑍̅̅̅̅ > 𝑋𝑌̅̅ ̅̅ , then the following mathematical statements are true.

𝑚∠𝑌> 𝑚∠𝑋

𝑚∠𝑋> 𝑚∠𝑍

𝑚∠𝑌> 𝑚∠𝑍

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What’s New

Mang Juan is fencing the triangular lot that he was able

to purchase. His available material is 11-meter barbed wire.

How much more barbed wire will he need to fence his lot?

What is It

ILLUSTRATIVE EXAMPLE 1

Use the Exterior Angle Inequality theorem to write the inequalities observable in the figure shown.

4 meters

7 meters

Based on the activity, how did you come up with the measurement of the third side of the triangular garden?

In this module, we will apply the triangle inequality theorems in different kinds of problem.

The measure of an exterior angle of a triangle is greater than the measure of any of its remote interior angles.

Exterior Angle Inequality Theorem

SOLUTION Consider ∆𝐻𝐴𝑇:

𝑚∠𝐻𝐴𝑀 > 𝑚∠𝐻𝑇𝐴

𝑚∠𝐻𝐴𝑀 > 𝑚∠𝐴𝐻𝑇

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ILLUSTRATIVE EXAMPLE 2

Given that the measure of the exterior angle is equal to the sum of the measures of its

remote interior angles, solve for 𝑥.

ILLUSTRATIVE EXAMPLE 1

In ∆𝑋𝑌𝑍, if 𝑋𝑍̅̅ ̅̅ > 𝑌𝑍̅̅̅̅ and 𝑌𝑍̅̅̅̅ > 𝑋𝑌̅̅ ̅̅ , then it follows that,

𝑚∠𝑌 > 𝑚∠𝑋 𝑚∠𝑋 > 𝑚∠𝑍 𝑚∠𝑌 > 𝑚∠𝑍

ILLUSTRATIVE EXAMPLE 2 Given ∆𝐺𝐻𝐼 whose perimeter is 50 units, what is the largest angle?

SOLUTION:

Since the longest side is 𝐻𝐼, therefore the angle opposite to it is the largest angle

which is 𝐺.

𝑚𝐴 + 𝑚𝐵 = 𝑚𝐴𝐶𝐷

(2𝑥 + 3)° + (3𝑥 + 2)° = 135°

2𝑥 + 3 + 3𝑥 + 2 = 135

5𝑥 + 5 = 135

5𝑥 = 135 − 5

5𝑥 = 130

5𝑥

5=

130

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𝑥 = 26°

First, solve for 𝑥. 𝐺𝐼 + 𝐻𝐼 + 𝐺𝐻 = 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟

2𝑥 + 1 + 4𝑥 − 2 + 𝑥 + 2 = 50

7𝑥 + 1 = 50

7𝑥 = 50 − 1

7𝑥 = 49

7𝑥

7=

49

7

𝑥 = 7 𝑢𝑛𝑖𝑡𝑠

Second, find the measure of each side.

𝐺𝐼 = 2𝑥 + 1 = 2(7) + 1 = 15𝑢𝑛𝑖𝑡𝑠

𝐻𝐼 = 4𝑥 − 2 = 4(7) − 2 = 26 𝑢𝑛𝑖𝑡𝑠

𝐺𝐻 = 𝑥 + 2 = 7 + 2 = 9 𝑢𝑛𝑖𝑡𝑠

If one side of a triangle is longer than the second side, then the angle

opposite the first side is larger than the angle opposite the second side.

Triangle Inequality Theorem 1 (Ss → Aa)

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ILLUSTRATIVE EXAMPLE 1

Name the shortest side and the longest side of the following triangles.

Triangle Longest Side Shortest Side

1 ∆𝐽𝐾𝐿 𝐽𝐾̅̅ ̅ 𝐽�̅�

2 ∆𝑀𝑁𝑂 𝑀𝑂̅̅ ̅̅ ̅ 𝑀𝑁̅̅ ̅̅ ̅

3 ∆𝑃𝑄𝑅 𝑃𝑅̅̅ ̅̅ 𝑄𝑅̅̅ ̅̅

ILLUSTRATIVE EXAMPLE 2 What is the shortest side of ∆𝐽𝐾𝐿?

To find the shortest side, solve first for 𝑥. 𝑚𝐽 + 𝑚𝐾 + 𝑚𝐿 = 180°

(3𝑥)° + (2𝑥 − 5)° + (𝑥 − 7)° = 180°

3𝑥 + 2𝑥 − 5 + 𝑥 − 7 = 180

6𝑥 − 12 = 180

6𝑥 = 180 + 12

6𝑥 = 192

6𝑥

6=

192

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𝑥 = 32°

Next, find the measure of each angle.

𝑚𝐽 = 3𝑥 = 3(32) = 96°

𝑚𝐾 = 2𝑥 − 5 = 2(32) − 5 = 59°

𝑚𝐿 = 𝑥 − 7 = 32 − 7 = 25°

Since the smallest angle is 𝐿 ,then the shortest

side is 𝐽𝐾.

If one angle of a triangle is larger than the second angle, then the side opposite the first angle is longer than the side opposite the second angle.

Triangle Inequality Theorem 2 (Aa → Ss)

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Triangle Inequality Theorem 3 (S1 + S2 > S3)

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ILLUSTRATIVE EXAMPLE If two sides of a triangle have lengths 7 cm and 10 cm, what are the possible lengths of the third side?

If t is the third side then, the following should be satisfied:

Inequality 1 Inequality 2 Inequality 3

7 + 10 > t

t < 7+10

t < 17

t must be less than 17

7 + t > 10

t > 10 − 7

t > 3

t must be greater than 3

t + 10 > 7 t > 7 − 10

t > -3

Nonpositive values of t to be disregarded.

The resulting inequalities show that t must be between 3 and 17. Side t may have the following measurements (in cm): {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}.

The inequality model representing the possible length of the third side is 𝟑 < 𝒕 < 𝟏𝟕.

ILLUSTRATIVE EXAMPLES Application 1. Using Hinge Theorem and its Converse, solve for the possible values of m.

SOLUTION:

2𝑚 − 1 > 𝑚 + 4 2𝑚 − 𝑚 > 1 + 4

𝑚 > 5

If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second.

Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second.

Converse of Hinge Theorem (SSS Triangle Inequality Theorem)

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Therefore, Adee is farther from the center of the plaza than Ella.

2. Ella and Adee enjoy biking every afternoon at the plaza. One day, they decided to go on

different directions. From the center of the plaza, Ella bikes 40 meters east and then 50

meters south. Adee bikes 50 meters west. He then takes a right of 70° and bikes 40 meters.

Who is farther from the center of the plaza?

SOLUTION:

What’s More

Directions: Write your answers on a separate sheet.

A. Given the following measurements of the sides of a triangle, list down its angles from smallest to largest.

1. 𝐴𝐵 = 17 , 𝐵𝐶 = 21, 𝐴𝐶 = 18

2. 𝐴𝐵 = 15 , 𝐴𝐶 = 16, 𝐵𝐶 = 17 B. Given the following measurements of the angles of a triangle, list down its sides from

shortest to longest.

1. 𝑚∠𝐴 = 460, 𝑚∠𝐵 = 300 , 𝑚∠𝐶 =? 2. 𝑚∠𝐶 = 1010, ∠𝐵 = 700 𝑚∠𝐴 =? 3. 𝑚∠𝐴 = 590 , 𝑚∠𝐶 = 610 𝑚∠𝐵 =?

C. Decide whether each set of numbers can form a triangle.

Given a + b > c a + c > b b + c > a Triangle? YES/NO

EXAMPLE: {5,8,10}

13>10 15>8 18>5 YES

1. {5, 5, 3}

2. {6, 2, 7}

3. {11, 8, 3}

4. {5, 5, 10}

5. {4, 5, 8}

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D. Using Hinge Theorem and Its Converse, write a conclusion about each statement. Use the symbols >, < or = to complete the statements about the figure shown below.

What I Have Learned

Directions: Briefly answer the questions below.

QUESTIONS RESPONSES

1. How would you compare the measure of the exterior angle and its remote interior angles?

2. Where would the largest angle of a triangle be located? (Justify your answer using the Triangle inequality theorem 1.)

3. Where would the shortest side of a triangle be located? (Justify your answer using Triangle Inequality Theorem 2)

4. Can the following side measures form a triangle? 2,2,10. Why?

5. In figure 1, ∠𝐷𝐴𝐶 is the exterior angle

while ∠𝐴𝐵𝐶 and ∠𝐴𝐶𝐵 are the remote interior angles. Describe the measurements of the exterior angle and the remote interior angles.

What I Can Do

Directions: Briefly answer the reflection questions below.

1. What is the importance of learning Triangle Inequality Theorems?

2. How can you apply the Triangle Inequality Theorems in real-life problems?

STATEMENTS

If 𝐴𝐶̅̅ ̅̅ ≅ 𝐴𝐷̅̅ ̅̅ 𝑎𝑛𝑑 𝑚∠1 ≅ 𝑚∠2 𝐵𝐶̅̅ ̅̅ 𝐵𝐷̅̅ ̅̅

If 𝐵𝐶̅̅ ̅̅ ≅ 𝐵𝐷̅̅ ̅̅ 𝑎𝑛𝑑 𝐴𝐶̅̅ ̅̅ ≅ 𝐴𝐷̅̅ ̅̅ 𝑚∠4 𝑚∠3

If 𝐴𝐷̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅ 𝑎𝑛𝑑 𝑚∠2 < 𝑚∠1 𝐵𝐷̅̅ ̅̅ 𝐵𝐶̅̅ ̅̅

If 𝐵𝐷̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ 𝑎𝑛𝑑 𝐴𝐷̅̅ ̅̅ > 𝐴𝐶̅̅ ̅̅ 𝑚∠3 𝑚∠4

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Assessment

Direction: Read each item carefully. Write the letter of the correct answer on a separate sheet. 1. Triangle inequality theorem 3 states that the sum of the lengths of any two sides of a

triangle is _______ the length of the third side.

A. greater than B. less than

C. equal to D. greater than but not equal to

2. Which of the following statements is correct?

A. The sum of the remote interior angle is equal to the measurement of the exterior angle

of the triangle.

B. The sum of the remote interior angle is greater than the measurement of the exterior

angle of the triangle.

C. The sum of the remote interior angle is less than the measurement of the exterior angle

of the triangle.

D. The measurements of the angles of an equilateral triangle are not equal.

3. The largest angle of a triangle is located across the _________ of the triangle.

A. from the smallest side B. from the longest side

C. adjacent to the smallest side D. adjacent to the longest side

4. Where would the longest side of a triangle be located?

A. Across from the smallest angle of a triangle

B. Adjacent to the largest side of a triangle

C. Across the largest angle of a triangle

D. In the middle of a triangle

5. Which of these lengths CAN represent the sides of a triangle?

I. 3 meters, 4 meters, 5 meters

II. 2 inches, 2 inches, 10 inches

III. 5 feet, 5 feet, 5 feet

IV. 10 meters, 3 meters, 2 meters

A. I, III, and IV B. I and II C. I and III D. III only

6. The exterior angle of a triangle measures 100o. if one of the remote interior angles of the

same triangle is 58o, what is the measurement of the other remote interior angle?

A. 42° B. 43° C. 45° D. 92°

7. What is the measurement of the exterior angle of an equiangular triangle?

A. 120° B. 100° C. 60° D. 30°

8. Which of the following does not represent the lengths of the sides of a triangle?

A. 2cm,2cm,1cm B. 3cm,3cm,15cm

C. 5cm,4cm,3cm D. 1cm,1cm,1cm

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9. Given the lengths of two sides of a triangle. Find the range of lengths of the third side: 13

cm and 7 cm.

A. 6 < 𝑥 < 20 B. 20 < 𝑥 < 6

C. 6 < 𝑥 > 20 D. 6 > 𝑥 > 20

10. Hinge Theorem states that” If two triangles have two congruent sides, then the one with the larger included angle has the ____________”.

A. longer third side B. shorter third side

C. largest angle D. shortest angle

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Answer Key

What’s In:

1.D 2. E 3. B 4. C 5. A

What’s New:

3 meters to 6 meters

What I Know:

1.B 2. D 3. D 4. A 5. A

What’s More:

A. B.

1. smallest- C 1. smallest- 𝑨𝑪̅̅̅̅ 3. smallest- 𝑩𝑪̅̅̅̅

larger - B longer - 𝑪𝑩̅̅̅̅ longer - 𝑨𝑪̅̅̅̅

largest - A longest - 𝑨𝑩̅̅̅̅ longest - 𝑨𝑩̅̅̅̅

2. smallest- C 2. smallest- 𝑩𝑪̅̅̅̅

larger - B longer - 𝑨𝑪̅̅̅̅

largest - A longest - 𝑨𝑩̅̅̅̅

What’s More:

C. D.

Given a + b > c a + c > b b + c > a Triangle? Yes/No

1 5,5,3 10> 8>5 8>5 YES

2 6,2,7 8>7 13>2 9>6 YES

3 11,8,3 19>3 14>8 11=11 NO

4 5,5,10 10=10 15>5 15>5 NO

5 4,5,8 9>8 12>5 13>4 YES

𝐵𝐶̅̅̅̅ = 𝐵𝐷̅̅̅̅

𝑚∠4 = 𝑚∠3

𝐵𝐷̅̅̅̅ < 𝐵𝐶̅̅̅̅

𝑚∠3 > 𝑚∠4

What I Have Learned:

(For Item 1-4, answers may vary.)

1.The measure of the exterior angle is greater than the measure of the remote interior

angle.

2.The largest angle is opposite the longest side.

3.The shortest side is opposite the smallest angle.

4.No, because one condition will not be satisfied (2+2<10).

5.𝑚𝐷𝐴𝐶>𝑚𝐴𝐵𝐶

𝑚𝐷𝐴𝐶>𝑚𝐴𝐶𝐵

What I Can Do: (Answers may vary.)

1.It can help us solve some real-life problems. 2.It can be used in architecture, engineering, fashion trends and the like.

13

Assessment:

1.A 3. B 5. C 7. A 9. A

2.A 4. C 6. A 8. B 10. A

14

References:

Abuzo, Emmanuel P. et. al., Mathematics Learner’s Manual 8, Pasig: DepEd- IMCS, 2013

373-433

15

I AM A FILIPINO

by Carlos P. Romulo

I am a Filipino – inheritor of a glorious past, hostage to the

uncertain future. As such, I must prove equal to a two-fold

task – the task of meeting my responsibility to the past, and

the task of performing my obligation to the future.

I am sprung from a hardy race – child many generations

removed of ancient Malayan pioneers. Across the centuries,

the memory comes rushing back to me: of brown-skinned

men putting out to sea in ships that were as frail as their hearts

were stout. Over the sea I see them come, borne upon the

billowing wave and the whistling wind, carried upon the

mighty swell of hope – hope in the free abundance of the new

land that was to be their home and their children’s forever.

This is the land they sought and found. Every inch of shore

that their eyes first set upon, every hill and mountain that

beckoned to them with a green and purple invitation, every

mile of rolling plain that their view encompassed, every river

and lake that promised a plentiful living and the fruitfulness

of commerce, is a hollowed spot to me.

By the strength of their hearts and hands, by every right of

law, human and divine, this land and all the appurtenances

thereof – the black and fertile soil, the seas and lakes and

rivers teeming with fish, the forests with their inexhaustible

wealth in wild and timber, the mountains with their bowels

swollen with minerals – the whole of this rich and happy land

has been for centuries without number, the land of my

fathers. This land I received in trust from them, and in trust

will pass it to my children, and so on until the world is no

more.

I am a Filipino. In my blood runs the immortal seed of heroes

– seed that flowered down the centuries in deeds of courage

and defiance. In my veins yet pulses the same hot blood that

sent Lapulapu to battle against the alien foe, that drove Diego

Silang and Dagohoy into rebellion against the foreign

oppressor.

That seed is immortal. It is the self-same seed that flowered

in the heart of Jose Rizal that morning in Bagumbayan when

a volley of shots put an end to all that was mortal of him and

made his spirit deathless forever; the same that flowered in

the hearts of Bonifacio in Balintawak, of Gregorio del Pilar

at Tirad Pass, of Antonio Luna at Calumpit, that bloomed in

flowers of frustration in the sad heart of Emilio Aguinaldo at

Palanan, and yet burst forth royally again in the proud heart

of Manuel L. Quezon when he stood at last on the threshold

of ancient Malacanang Palace, in the symbolic act of

possession and racial vindication. The seed I bear within me

is an immortal seed.

It is the mark of my manhood, the symbol of my dignity as

a human being. Like the seeds that were once buried in the

tomb of Tutankhamen many thousands of years ago, it shall

grow and flower and bear fruit again. It is the insigne of my

race, and my generation is but a stage in the unending

search of my people for freedom and happiness.

I am a Filipino, child of the marriage of the East and the

West. The East, with its languor and mysticism, its passivity

and endurance, was my mother, and my sire was the West

that came thundering across the seas with the Cross and

Sword and the Machine. I am of the East, an eager

participant in its struggles for liberation from the imperialist

yoke. But I know also that the East must awake from its

centuried sleep, shake off the lethargy that has bound its

limbs, and start moving where destiny awaits.

For I, too, am of the West, and the vigorous peoples of the

West have destroyed forever the peace and quiet that once

were ours. I can no longer live, a being apart from those

whose world now trembles to the roar of bomb and cannon

shot. For no man and no nation is an island, but a part of the

main, and there is no longer any East and West – only

individuals and nations making those momentous choices

that are the hinges upon which history revolves. At the

vanguard of progress in this part of the world I stand – a

forlorn figure in the eyes of some, but not one defeated and

lost. For through the thick, interlacing branches of habit and

custom above me I have seen the light of the sun, and I

know that it is good. I have seen the light of justice and

equality and freedom, my heart has been lifted by the vision

of democracy, and I shall not rest until my land and my

people shall have been blessed by these, beyond the power

of any man or nation to subvert or destroy.

I am a Filipino, and this is my inheritance. What pledge

shall I give that I may prove worthy of my inheritance? I

shall give the pledge that has come ringing down the

corridors of the centuries, and it shall be compounded of the

joyous cries of my Malayan forebears when first they saw

the contours of this land loom before their eyes, of the battle

cries that have resounded in every field of combat from

Mactan to Tirad Pass, of the voices of my people when they

sing:

“I am a Filipino born to freedom, and I shall not rest until

freedom shall have been added unto my inheritance—for

myself and my children and my children’s children—

forever.”