4 th QUARTER Module 4: PROVING PROPERTIES OF PARALLEL LINES CUT BY A TRANSVERSAL

Republic of the Philippines

Department of Education

Regional Office IX, Zamboanga Peninsula

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est for

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rogress

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eal of

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artnership

8

4

th

QUARTER – Module 4:

PROVING PROPERTIES OF PARALLEL

LINES CUT BY A TRANSVERSAL

Name of Learner:

___________________________

Grade & Section:

___________________________

1

Mathematics – Grade 8 Alternative Delivery Mode

Quarter 4 - Module 4:

Proving Properties of Parallel Lines cut by a Transversal

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: [email protected]

Development Team of the Module

Writer: Merry-Joy L. Caluyo

Editors: Loraine S. Enguio

Viola I. Quiniquito

Layout Artist: Abdurauf J. Baldomero

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

Principal Garry 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.

2

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 learn and understand the

concepts of proving properties of parallel lines cut by a transversal and polish your

reasoning skills. This was designed to address the needs of the student in learning

the Most Essential Learning Competencies for Grade 8 Mathematics. This module

was written with detailed definition of the important terms followed by illustrative

examples and activities.

After going through this module, you are expected to prove properties of parallel

lines cut by a transversal (M8GE-IVd-1).

3

What I Know

Directions: Read and understand each statement carefully. Write the letter of the

correct answer on a separate sheet.

1. All of the following are properties of parallel lines cut by a transversal EXCEPT;

a. Opposite sides are not parallel.

b. Alternate interior angles are congruent.

c. Corresponding angles are congruent.

d. Same-side interior angles are supplementary.

2. In Figure 1, 𝑎 ∥ 𝑏 and t is the transversal. What is the name of the angle formed

by ∠1 𝑎𝑛𝑑 ∠5?

a. Alternate angles

b. Corresponding angles

c. Consecutive angles

d. Vertical angles

3. In Figure 1, What is the relation between ∠2 𝑎𝑛𝑑 ∠8?

a. Alternate Exterior angles

b. Alternate Interior angles

c. Corresponding angles

d. Vertical Angles

4. In figure 2, if

𝑎 ∥ 𝑏 and t is a transversal, what is the relation exist

between ∠1 𝑎𝑛𝑑 ∠7?

a. Alternate Exterior angle

b. Corresponding angles

c. Same- side Interior angle

d. Same-side Exterior angle

5. All the statements below define parallel lines, EXCEPT:

a. Lines are on the same plane.

b. It has the same distance apart.

c. Lines that never touch.

4

What’s In

A. Directions: Match the different properties of parallel lines cut by transversal in

Column A with its definition in Column B. Write the letter of the correct answer on

a separate sheet.

COLUMN A

COLUMN B

1. Alternate Exterior Angles

2. Alternate Interior Angles

3. Same- Side Exterior Angles

4. Corresponding Angles

5. Same-side Interior Angles

a. Angles in similar position on the same side of

the transversal and are congruent.

b. Angles outside the two intersected lines on

opposite sides of the transversal and are not

adjacent angles.

c. Angles between the two intersected lines on

opposite sides of the transversal.

d. Angles between the two intersected lines in

the same side of the transversal.

e. Angles outside the two intersected lines in the

side of the transversal.

B. Directions: Given the figure below, 𝑐 ∥ 𝑑 and t is the transversal. Determine the

relationship between the following pairs of angles. Write your answers on a separate

sheet.

1.

∠1 𝑎𝑛𝑑 ∠3

____________________________

2.

∠4 𝑎𝑛𝑑 ∠5

____________________________

3.

∠6 𝑎𝑛𝑑 ∠7

____________________________

4.

∠3 𝑎𝑛𝑑 ∠6

____________________________

5.

∠1 𝑎𝑛𝑑 ∠8

____________________________

5

Directions: Read and understand the statement below and write your answer on a

separate sheet.

Lines a and b are parallel cut by transversal t, which is

also perpendicular to a and b. Which statement is NOT

correct? Justify your answer.

a.

∠1 𝑎𝑛𝑑 ∠6 𝑎𝑟𝑒 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡

b.

∠2 𝑎𝑛𝑑 ∠3 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦

c.

∠3 𝑎𝑛𝑑 ∠5 𝑎𝑟𝑒 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑎𝑛𝑔𝑙𝑒𝑠

d.

∠1 𝑎𝑛𝑑 ∠4 𝑓𝑜𝑟𝑚𝑠 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑝𝑎𝑖𝑟

Lesson

Properties of Parallel Lines

cut by a Transversal

What is It

Parallel lines are lines on the same plane, the same distance apart and never

touches or intersect.

Transversal line is a line that intersects two or more lines.

m ║ n read as “m is parallel to n” and t is a transversal.

6

If two parallel lines are cut by a transversal, then the sum of the measures of

the interior angles on the same side of the transversal is

180⁰. And the sum of the

measures of the exterior angles on the same side of the transversal is also 180⁰

Proofs can be written in 3- different ways:

Given: 𝑝 ∥ 𝑞

Prove: ∠4 𝑎𝑛𝑑 ∠6 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦

Proof:

Since

𝑝 ∥ 𝑞

, then

∠1 ≅ ∠5

by definition of corresponding angles. Thus,

∠3 ≅ ∠6

by definition of alternate interior angles. So,

∠4 𝑎𝑛𝑑 ∠6 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦

by definition

of same side interior angles.

STATEMENTS

REASONS

𝑝 ∥ 𝑞

Given

∠3 &∠4 form a linear pair

Definition of linear pair

𝑚∠3 + 𝑚∠4 = 180°

Sum of angles of a linear pair is equal to 180°

∠3 ≅ ∠6

Alternate interior angles are congruent

𝑚∠3 = 𝑚∠6

Definition of congruent angles

𝑚∠6 + 𝑚∠4 = 180°

Substitution Property of Equality

∠4 𝑎𝑛𝑑 ∠6 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦

Definition of Supplementary Angles

Flow Chart Proof

𝑝 ∥ 𝑞 Given ∠1 ≅ ∠5 Corresponding angles are congruent ∠3 ≅ ∠6 Alternate Interior angles are congruent ∠4 𝑎𝑛𝑑 ∠6 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 Same side interior angles are

supplementary

Paragraph Form

7

Find the value of x to show that

𝑎 ∥ 𝑏 given the following:

1 6 7 2 4 8 5 a b 3

t

Thus, 𝒙 = 𝟓𝟎° and alternate interior angles are

congruent.

∴ 𝑎 ∥ 𝑏

Given:

𝒎∠𝟏 = (𝟐𝒙 + 𝟐𝟓)° 𝒂𝒏𝒅 𝒎∠𝟖 = (𝒙 + 𝟕𝟓)°

Solution:

Let, 𝑚∠1 = 𝑚∠8

2𝑥 + 25 = 𝑥 + 75

2𝑥 − 𝑥 = 75 − 25

𝒙 = 𝟓𝟎°

𝒎∠𝟏 = 𝟐𝒙 + 𝟐𝟓 𝒎∠𝟖 = 𝒙 + 𝟕𝟓

𝑚∠1 = 2(50) + 25 𝑚∠8 = (50) + 75

𝑚∠1 = 125° 𝑚∠8 = 125°

Using Algebra to show two lines are parallel

Example 1 Example 2

Given:

𝑚∠𝟑 = (𝟒𝒙 − 𝟒𝟏)° 𝒂𝒏𝒅 𝒎∠𝟕 = (𝟐𝒙 + 𝟕)°

Solution:

Let, 𝑚∠3 = 𝑚∠7

4𝑥 − 41 = 2𝑥 + 7

4𝑥 − 2𝑥 = 7 + 41

2𝑥 2

=

48 2

𝐱 = 𝟐𝟒°

𝒎∠𝟑 = 𝟒𝒙 − 𝟒𝟏 𝒎∠𝟕 = 𝟐𝒙 + 𝟕

𝑚∠3 = 4(24) − 41 𝑚∠7 = 2(24) + 7

𝑚∠3 = 96 − 41 𝑚∠7 = 48 + 7

𝑚∠3 = 55° 𝑚∠7 = 55°

Thus, 𝒙 = 𝟐𝟒° and corresponding angles are congruent.

8

What’s More

Directions: Prove that two lines are parallel by supplying the missing statement and

reason given below.

Given:

x ∥ y and t is a transversal

Prove:

∠2 ≅ ∠6

Proof:

STATEMENTS

REASONS

𝑥 ∥ 𝑦 and t is a transversal

Given

∠2 ≅ ∠7 form a linear pair

Definition of a linear pair

1.)

Sum of angles of a linear pair is equal to 180°

𝑚∠7 + 𝑚∠6 = 180°

2.)

𝑚∠2 + 𝑚∠7 = 𝑚∠7 + 𝑚∠6

3.)

4.)

Addition Property of Equality

∠2 ≅ ∠6

5.)

Directions: Find the value of x to show that

𝑎 ∥ 𝑏. Show your solutions.

TERM

ACTIVITY 1

1 6 7 2 4 8 5 a b 3

t

Given:

m∠2 = (2x − 5)° and m∠6 = (x + 25)°

Solution:

TERM

ACTIVITY 2

2 x y t 1 8 7 3 6 4 5

9

What I Have Learned

Directions: To test what you have learned from this lesson, Supply the reason to the

following statement using the different properties of parallel lines cut by a transversal.

STATEMENTS

REASONS

∠2 ≅ ∠7

1.

𝑚∠2 + 𝑚∠8 = 180°

2.

𝑚∠1 = 𝑚∠5

3.

𝑚∠3 + 𝑚∠5 = 180°

4.

∠4 ≅ ∠5

5.

What I Can Do

Directions: Determine whether the underlined word/phrase from the given statement

implies two lines are parallel. Write Parallel if it does and NOT if it does not on a

separate sheet.

1. Pedestrian lanes is used to cross the road.

2. The car wipers is used to have clear visibility while driving.

3. Nena removes the dust on the opposite sides of a picture frame.

4. After researching, Roy arranges the books on the shelf.

5. The boy uses ladder to pick mangoes from the tree.

n m p

10

Assessment

Directions: Complete the proof table by picking the statement or reason given below.

Given: 𝑝 ∥ 𝑞

Prove: ∠2 𝑎𝑛𝑑∠3 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦

Proof:

STATEMENTS

REASONS

1.)

Given

∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦

2.)

𝑚∠1 + 𝑚∠2 = 180°

3.)

4.)

5.)

6.)

7.)

8.)

9.)

10)

Definition of supplementary angles

p ∥ q

Linear Pair Postulate

∠2 𝑎𝑛𝑑∠3 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦

If two parallel lines are cut by a transversal, then the corresponding angles are congruent.

Definition of congruent angles

Substitution property

Definition of supplementary angles

∠1 ≅ ∠3 𝑚∠1 = 𝑚∠3

11

Answer Key

Wha t I Know : 1. A 3 . A 5 . D 2. B 4 . D 3. Wha t’s In : A. B. 1. b 1. C orr es po nd in g a ng le s 2. c 2. A lte rn ate e xte rio r a ng le s 3. e 3. S am e s id e i nte rio r a ng le s 4. a 4. A lte rn ate in te rio r a ng le s 5. d 5. A lte rn ate e xte rio r a ng le s Wha t’s N ew : An sw er: D, b ec au se ∠1 𝑎 𝑛𝑑 ∠4 a re n ot ad ja ce nt an gle s, th ere fo re th ey a re n ot a lin ea r p air . Wha t’s M ore: Act ivity 1

STA

TEM

ENT

REA

SON

𝑥∥

𝑦

Give

n

∠2≅

∠7 f

orm

a l

in

ea

r p

air

De

fin

iti

on

o

f a

lin

ea

r p

air

1.)

𝑚∠2

+𝑚

∠7=

180°

Sum

of

an

gle

s o

f a

lin

ea

r p

air

is

eq

ua

l t

o

180°

𝑚∠7

+𝑚

∠6=

180°

Same

-si

de

in

te

rio

r a

ng

le

s a

re

su

pp

le

men

ta

ry

𝑚∠2

+𝑚

∠7=

𝑚∠7

+𝑚

∠6

Sub

stit

uti

on

Pro

pe

rty_

4)

𝑚∠2

=𝑚

∠6

Add

iti

on

Pro

pe

rty

of

Equ

alit

y

∠2≅

∠6

De

fin

iti

on

o

f co

ng

rue

nt

an

gle

s

12 Wha t’s M ore: Ac tiv ity 2

Giv

en

:

m∠2

=(

2x−

5)°

an

d m∠

6=

(x

+25

)°

Solu

tio

n

Let , 𝑚∠2 =𝑚 ∠6 2𝑥 −5 =𝑥 +25 2𝑥 −𝑥 =25 +5 𝑥= 30° 𝑚∠2 =2 𝑥− 5 𝑚∠6 =𝑥 +25 𝑚∠2 =2 (30 )− 5 𝑚∠6 =30 +25 𝑚∠2 =60 −5 𝑚∠6 =55° 𝑚∠2 =55° Th us, 𝑥= 30° a nd co rre sp on din g a ng les ar e c on gru en t ∴

𝑎∥

𝑏

Wha t I Have Lear ned : 1. Alt ern ate E xte rio r a ng le s a re c on gru en t. 2. Sa me -s id e e xte rio r a ng le s a re s up ple me nta ry . 3. Co rre sp on din g a ng le s a re c on gru en t. 4. Sa me -s id e i nte rio r a ng le s a re c on gru en t. 5. Alt ern ate in te rio r a ng le s a re c on gru en t. Wha t I Ca n D o:

1.

Para

lle

l

2

. Pa

ralle

l

3

. Pa

ralle

l

4

. Pa

ralle

l

5

. N

ot

13 Ass ess men t: AS SE SS ME NT 1

STA

TEMEN

TS

REA

SON

S

1.

p∥

q

Give

n

∠1 a

nd

∠2

are

sup

ple

men

tary

Lin

ea

r Pa

ir

Post

ula

te

m∠1

+m∠

2=

180°

De

fin

iti

on

o

f su

pp

le

me

nta

ry a

ng

le

4.)

∠1≅

∠3

If

tw

o

pa

ralle

l

lin

es

are

cut

by

a

tra

nsve

rsal,

th

en

th

e

corre

sp

on

din

g

an

gle

s a

re co

ng

rue

nt.

6.)

m∠1

=m∠

3

De

fin

iti

on

o

f co

ng

rue

nt

an

gle

s.

8.)

m∠1

+m∠

3=

180°

Sub

stit

uti

on

p

rop

ert

y

10)

∠2 a

nd

∠3

are

sup

ple

men

tary

De

fin

iti

on

o

f su

pp

le

me

nta

ry a

ng

le

s

14

References:

Abuzo, E., et al. (Mathematics Learner’s Module (First Edition), Pasig; DepEd-IMCS,

2013), pp. 452-462.

Dilao, S.J., & Bernabe, G.(GEOMETRY III (Revised edition),Quezon; SD publications,

2009), pp.77-78

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.”