Mathematics 8 Module, Lecture notes of Mathematics

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Department of Education • Republic of the Philippines

Mathematics

Quarter 3 – Module 1:

Week 1 – Week 4

Mathematics – Grade 8

Alternative Delivery Mode

Quarter 3 – Module 1: Week 1- 4

First Edition, 2020

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Published by the Department of Education

Secretary:

Undersecretary:

Assistant Secretary:

Printed in the Philippines by ________________________

Department of Education – Bureau of Learning Resources (DepEd-BLR)

Office Address: ____________________________________________

____________________________________________

Telefax: ____________________________________________

E-mail Address: ____________________________________________

Development Team of the Module

Authors: : Flora S. Isada , Madilane A.Malla, Cherry F.Clerigo, Daryl T.Dela Cruz,

Florence C. Soriano , Caren Lynn N. Diaz , Cheryl B. Caliguiran, Ma. Cristina

Getigan, Lorelie R. Huidem , Beatrice Tarcena , Miriam F. Genoza ,Reggie D. Manzano,

Hilda S. Fabriaga ,

Editor:

Reviewers:

Illustrator:

Layout Artist:

Management Team:

CONTENT STANDARDS : The learner demonstrates understanding of key concepts of

inequalities in a triangle, and parallel and perpendicular lines

PERFORMANCE STANDARDS : The learner is able to communicate mathematical thinking

with coherence and clarity in formulating, investigating, analyzing, and solving real-life

problems involving triangle inequalities, and parallelism and perpendicularity of lines using

appropriate and accurate representations.

The module is divided into four lessons, namely:

• Lesson 1 – Mathematical System
• Lesson 2 – Defined and Undefined Terms
• Lesson 3 – Postulates and Theorems
• Lesson 4 – Triangle Congruence
• Lesson 5 – Triangle Congruence Postulates

After going through this module, you are expected to:

• Describe a mathematical system
• Illustrates the need for an axiomatic structure of a mathematical system in general, and

in Geometry in particular: (a) defined term (b) undefined terms;

• Apply the Theorems in Geometry.
• Define point, line, and plane.
• Identify and describe point, line and plane.
• Define point, line, and plane.
• Identify and describe point, line and plane.
• State the SAS Congruence Postulate.
• Illustrate the SAS Congruence Postulate.
• Illustrate ASA and AAS triangle congruence.
• Apply deductive skills to show congruence between two triangles.
• Illustrates SSS Congruence Postulates
• Prove the congruence of triangles using SSS Postulate

Week 1 ( Day 1 – 2 )

What I Know

Directions: Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper.

1. Which of the following strongly describes mathematical system?

A. Consists of some undefined terms (primitive terms).

B. A statement that is assumed to be true

C. Has four parts: undefined terms, defined terms, postulates, theorem

D. none of these

2. Points or objects that are in the same plane.

A. collinear B. non-collinear C. coplanar D. non-coplanar

3. Which is part of a line that has one endpoint and continues on forever in the other direction?

A. Line B. Ray C. line segment D. opposite rays

4. What one shows a line segment?

A. B. C. D.

5.. Find MQ and PQ.

A. MQ=4, PQ=8 B. MQ=4, PQ=4 C. MQ=8, PQ=4 D. MQ=8, PQ=

A Mathematical system is a structure formed from one or more sets of undefined objects, various concepts

which may or may not be defined, and a set of axioms relating these objects and concepts.

What are the four parts of mathematical system?

➢ Structure of Mathematical Systems Mathematics can be divided into four major areas- higher

arithmetic, algebra, geometry, and analysis.

Generally., there are two elements that compose a mathematical system – vocabulary and principles.

➢ Vocabulary refers to either undefined terms or definition.

The four components of a mathematical system are as follows.

➢ Defined terms.

➢ Undefined terms.

➢ Theorems.

➢ Axioms and postulates..

Undefined Terms

Undefined terms are terms that have not been previously categorically determined. These terms are required

to establish the definition of other terms. Otherwise the system will be confined to a cyclic inquiry

“ What is the meaning of…”

Geometry, as a mathematical system , is founded by the following undefined terms:

1. points
2. lines
3. planes

Point – is a fixed location in space. It has no size, but it can be modeled by a dot.

The point shown below is called C or point C.

•C

Line – is a set of points. It is straight. It has infinite length but no thickness.

The line shown below is called line m or line AB or line AB or (AB) ̅ or (BA) ̅

𝐴𝐵

⃡

Plane – is a set of points extending in all directions. It has no length and width but no thickness.

The plane shown below can be named plane R.

Lesson 1 Mathematical System

M is the midpoint of A and B since AM = MB

➢ Ray is a part of a line. It has one endpoint but extends infinitely in one direction.

Note: In naming a ray, start first with the endpoint.

Example:

B is the midpoint of AC. If AB=2x+1 and BC = 3x-4 what is x? What is the measure of BC? What is the

measure of AC?

Solution:

the measure of AB equals the measure of BC

2x+1=3x- 4

4+1=3x-2x

5=x

BC=3x-4=3(5)-4=15-4=

AB=2x+1=2(5)+1=10+1=

AC=AB+BC=11+11=

➢ Opposite ray are rays with common endpoint but extending in opposite directions

The two opposite rays here are

𝐵𝐴 and 𝐵𝐶

Example1:

In a figure (a), which points are collinear?

In figure (b), which points are coplanar?

Answer:

a.) Points M, A, and T are collinear. Points B, A, N, and D are

also collinear. However points B, A and T are non-collinear.

b.)Points H, G and K are coplanar. Points H, G, K and L are non-coplanar.

Example 2:

Name all the segment, rays and opposite rays in the figure below.

Answer:

The line segments are 𝐴𝐵

and ̅𝐷𝐵

The rays are 𝐴𝐵 , 𝐵𝐶 , 𝐶𝐴 , 𝐵𝐴 and 𝐵𝐷.

The opposite rays are 𝐵𝐴 and 𝐵𝐶

Activity1:

Draw a HAPPY FACE if the statement is true. Otherwise draw a SAD FACE

1. Points A, B and F are collinear.
2. Points E, F and A are coplanar.
3. Points C, B and A are non-coplanar.
4. Points A, D and B are non-collinear.
5. Points F, E and C are coplanar.

Activity 2:

Write TRUE if the statement is true. Otherwise write FALSE.

1. A ray has definite length.
2. A line segment has definite length.

3. The endpoint of 𝑆𝑅 is S.

4. Opposite Rays are coplanar.

5. If 𝑌𝑋 and 𝑌𝑍 have a common endpoint, then they are opposite rays.

Activity 3:

Solve:

1. B is the midpoint of AC. Find AB.

2. Find KL.

3. Solve for x

WHAT I HAVE LEARNED

Fill in the blanks:

1. The Four components of mathematical system are _______________, _______________ , ________________

and _______________.

2. ____________________ are points that lie on the same line.
3. A ____________ is a part of a line that has two endpoints.
4. points that lie on the same plane are _______________ points.
5. A part of a line that has one endpoint is a _______
6. If B is the midpoint of 𝐴𝐶

, then _______

7. If O is between 𝑇𝑌

, then _______

ASSESSMENT

A. Choose the letter of the correct answer:

1. 𝐸𝐴 and 𝐸𝐵 are opposite rays

a. true

b. false

2. If a point is between two other points, what shape will the resulting line segment have?

a. zig zag c. straight

b. curved d. angled

3. If E is the midpoint of 𝐵𝐷

, then

a. BD = DE c. BE < ED

b. BE = ED d. BD > DE

4. Part of a line. Has two endpoints and includes all of the points in between.

a. point c. line segment

b. line d. angle

5. What is the value of x?

a. x = 3 c. x = 1

b. x = 2 d. x = 5

Plane – is a set of points that extends without ends in all directions along a flat surface. A plane has no

thickness no edged; it is usually shown as four-sided shape.

Collinear points – are points that lie on the same line

Coplanar points – are point that lie in the same plane

Line segment – is a set of points and has a specific length. Its length is finite and is determined by its two

endpoints. In symbol 𝐴𝐵

Ray – any of a set of straight lines passing through one point. In symbol 𝐴𝐵

We can summarize Undefined terms as;

ACTIVITY I. (Hidden Word)

Match column A to column B to find out the hidden word.

Column A Column B

1. A set of all points E. Line

2. A flat surface T. Point

3. Points that lie on the same plane H. Space

4. Has length, but no width or thickness O. Plane

5. Points that lie on the same line N. Coplanar

6. Occupies no space, no width or length S. Collinear

WHAT IS IT

UNDEFINED TERMS (Naming the Figures)

So here we are going to nae undefined term through figures. Study them carefully.

Case 1: Sketch the following:

1. Point C on a plane S.
2. Opposite rays 𝐺𝐴 , 𝐺𝐸
3. Line LM.

Case 2: Use the figure below to answer the following:

A. Give the five other names for 𝑙𝑖𝑛𝑒 𝐶𝐺. 𝑜𝑟 𝐶𝐺

B. Give three points that are collinear in the

figure.

C. Give two names for Plane show.

D. Name the intersection in the plane and 𝑙𝑖𝑛𝑒 𝑓.

Observations:

Point is described as a dimensionless location in space, when you place two points on a plane, you can create

a line. Line described as set of all collinear points between and extending beyond two given points, and Plane

has two dimensions. It has infinitely length, infinitely width and zero height.

S

A

\

E

G

L M

Activity I.

Direction: Refer to the figure below and decide whether the statement is True (T) or False (F)

____ 1. Points 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶 are collinear.

____ 2. Points 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶 are coplanar.

____ 3. Point 𝐹 𝑙𝑖𝑒𝑠 𝑜𝑛 𝐷𝐸.

____ 4. 𝐷𝐸

____ 5. 𝐵𝐷

____ 6. 𝐵𝐷

ASSESSMENT

I. Multiple Choice: Choose the letter of the best answer. Write the chosen letter on a separate sheet

of paper.

1. a line is named by:

A. any two points on the line, on a lowercase script letter.

B. any one point on the line.

C. any three point on the line.

D. any three on the line on a uppercase script letter

2. A plane is named by.

A. any one point on the plane.

B. any three noncollinear points on the plane on an uppercase script letter.

C. any three collinear points on the plane on an uppercase script letter.

D. all points on the plane that create part of a line.

3.It is an exact location in space with no length or width.

A. Ray B. Line C. Point D. Line segment

4.Which of the following symbol must write in letter 𝐴𝐵. to name as 𝑙𝑖𝑛𝑒 𝐴𝐵?

A.

C. 𝐷.

5.Which of the following represents a point?

A. Strand of hair B. Bull’s eye in a target C. Book cover D. Flagpole

II. Direction: Use the figure below to answer the following:

1. Name three (3) points that are collinear.

2. Name four (4) points that are coplanar

3. Name three (3) points that are not collinear.

Choice: Multiple

Example 2 : 𝑚∠𝐴𝐷𝑆 = 30 and 𝑚∠𝑆𝐷𝐹 = 45 , what is 𝑚∠𝐴𝐷𝐹?

Applying Angle Addition Postulate,

𝑚∠𝐴𝐷𝐹 = 30 + 45 Substitute

the given

Example 3 : What is 𝑚∠𝑆𝐷𝐹, if 𝑚∠𝐴𝐷𝑆 = 34 and 𝑚∠𝐴𝐷𝐹 = 87?

Applying the Angle Addition Postulate

87 = 34 + 𝑚∠𝑆𝐷𝐹 Substitute the given

87 − 34 = 𝑚∠𝑆𝐷𝐹 Rewrite 34 on the left side of the equation and change its sign from positive to negative

𝟓𝟑 = 𝒎∠𝑺𝑫𝑭 or 𝒎∠𝑺𝑫𝑭 = 𝟓𝟑

➢ Supplement Postulate

If two angles form a linear pair, then they are supplementary

Example 1 :

∠𝐵𝑁𝐶 and ∠𝐶𝑁𝑀 form a linear pair. Therefore, ∠𝐵𝑁𝐶 and ∠𝐶𝑁𝑀 are supplementary. In symbols, we can

express this as:

Example 2 : If 𝑚∠𝑇𝑈𝑌 = 57 , what is 𝑚∠𝑌𝑈𝐼?

∠𝑇𝑈𝑌 and ∠𝑌𝑈𝐼 form a linear pair and they are supplementary.

57 + 𝑚∠𝑌𝑈𝐼 = 180 Substitute the given

𝑚∠𝑌𝑈𝐼 = 180 − 57 Rewrite 57 on the right side of the equation and change its sign from positive to

negative.

Now, let us summarize everything that we have discussed by completing the following statements:

• Ruler Postulate states that the distance between two distinct points of a line is the _______________ of the

difference of their coordinates.

• In Angle Addition Postulate, the measure of the _______________ angle is the sum of the two

_______________ angles formed when the _______________ point is connected to the vertex of the original

angle.

• Supplement Postulate states that two angles forming a _______________ are _______________.

Solve the given problem.

1. 𝐺 is an interior point of ∠𝐷𝐹𝐻, if 𝑚∠𝐷𝐹𝐺 = 19 and 𝑚∠𝐺𝐹𝐻 = 28 what is 𝑚∠𝐷𝐹𝐻?
2. ∠𝐵𝑁𝑀 and ∠𝑀𝑁𝑃 form a linear pair. If 𝑚∠𝐵𝑁𝑀 = 126 , what is 𝑚∠𝑀𝑁𝑃?

S

A

D

F

Linear Pair is composed of two adjacent angles whose two sides other than their common side

are opposite rays.

B

N

M

C

I

U

T

Y

ADDITIONAL ACTIVITY

Using the picture below, determine the distance between the given places.

1. School to Bookshop
2. School to Public Library
3. Public Library to Bookshop
4. Bookshop to School

ANSWER KEY

Week 2 ( Day 3 - 4)

Draw a ✓ if the given statement is TRUE , otherwise draw an x.

1. Vertical angles are congruent.
2. Supplements of congruent angles are congruent.
3. Complements of congruent angles are congruent.
4. Right angles are complementary.
5. Two angles are congruent if they have the same measure.
6. The sum of the angles in a triangle is 360°.
7. Two perpendicular lines form two right angles.
8. An isosceles triangle has three congruent sides.
9. If two sides of a triangle are congruent, then the angles opposite them are supplementary
10. If two angles of a triangle are congruent then the sides opposite them are congruent.

In Geometry, an ANGLE is the union of two non-collinear rays with a common endpoint called the VERTEX.

Angles could be acute, obtuse or right. Two angles are said to be CONGRUENT if they have EQUAL measures.

This Photo by Unknown Author is licensed under CC BY-NC-ND

10

- 5

2

To summarize everything that was discussed in this module, complete the following statements:

• According to the Vertical Angle Theorem, vertical angles are _______________.
• Isosceles Triangle Theorem states that if two sides of a triangle are congruent, then the angles

________________ them are _______________.

• In Isosceles Triangle Theorem, if two angles of a triangle are congruent, then the _______________ opposite

them are _______________.

All right triangles are ________________.

ADDITIONAL ACTIVITY

In this activity, you will try to find different objects around you that can help you illustrate the different

theorems that were discussed today. This may be anything that is found around you like those below:

You may take a picture of an object or cut pictures from a magazine or a newspaper. After choosing an object,

identify which specific part corresponds to the theorem and mark the congruent parts as shown below. Also,

specify the theorem that was illustrated on the picture.

Isosceles Triangle Theorem or

Converse of Isosceles Triangle Theorem Vertical Angle Theorem

ANSWER KEY

Two triangles are congruent if and only if their vertices can be paired so that

corresponding sides are congruent and corresponding angles are congruent.

Properties of Congruence

✓ Reflexive Property of Congruence

∠A ≅ ∠A ; AB

≅ AB

✓ Symmetric Property of Congruence

∠A ≅ ∠B , then ∠B ≅ ∠A

AB

≅ BC

, then BC

≅ AB

✓ Transitive Property of Congruence

A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C

AB

≅ BC

and BC

≅ CD

, then AB

≅ CD

Week 3 ( Day 1 - 2)

✓ Triangle is a 3-sided polygon or 2-dimensional figure with three sides, three angles and three vertices.

The sum of the three interior angles of a triangle is always 180°.

∆ symbol for triangle

✓ Congruency is a term used to describe two objects with the same shape and size.

≅ symbol for congruency

Have you ever wondered how bridges and buildings are designed? What factors are being

considered in the construction of buildings and bridges?

If ∆RUN ≅ ∆WOK (read as “ triangle RUN is congruent to triangle WOK’’ ), the vertices of the two triangles

correspond in the same order as the letters naming the triangles.

The correspondence among vertices can be used to name the corresponding congruent sides and angles

of the two triangles.

Example:

Given: ∆BAC ≅ ∆EDF

CONGRUENT SIDES CONGRUENT ANGLES

RU

≅ WO

∠R ≅ ∠W

UN

≅ OK

∠U ≅ ∠O

RN

≅ WK

∠N ≅ ∠K

R ↔ W

U ↔ O

N ↔ K

What do we mean when we say that

two triangles are congruent?

The symbol ↔ is read as

“ corresponds to ”

B. Directions: Complete each congruence statement by naming the congruent triangles,

corresponding angle or side.

1. ∆CAR ≅ ________ 2.∠B≅ ________

3. ________ ≅ BD

4. DE

≅ ________

5. ________ ≅ ∠PQM

C. Directions: Mark the angles and sides of each pair of triangles to indicate that they are congruent.

1. ∆CAB ≅ ∆FDE 2. ∆HIJ ≅ ∆STJ

3. ∆ZYX ≅ ∆ZCX 4. ∆IJK ≅ ∆DJC

5. ∆HGK ≅ ∆JKG

Week 3 ( Day 3 - 4)

Multiple Choice: Choose the letter of the best answer.

1. In ∆𝐴𝐵𝐶, 𝐴𝐵 = 𝐴𝐶. If 𝑚∠𝐵 = 80 , find the measure of ∠𝐴.

a. 20 b. 80 c. 100 d. 180

2. In ∆𝑀𝐴𝑅, what angle is included between 𝑀𝐴

and 𝐴𝑅

a. ∠𝑀 b. ∠𝐴 c. ∠𝐿 d. ∠𝑅

3. What property of congruence is illustrated in the statement? If 𝐴𝐵

and 𝐸𝐹

, then

a. Symmetric b. Transitive c. Reflexive d. Multiplication

4. Name the corresponding congruent parts as marked that will make each pair of triangles

congruent by SAS.

a. 𝐵𝑌

b. 𝐵𝑂

c. 𝑌𝑂

d. ∠𝐵 ≅ ∠𝑁, 𝐵𝑂

5. If the pictured triangles are congruent, what reason can be given?

a. SAS Postulate c. SSS Postulate

b. ASA Postulate d. the triangles are not necessarily congruent

In the previous lesson, you say that two triangles are congruent if their six corresponding

parts are congruent. However, you can have congruent triangles without having all corresponding

parts of the triangle congruent. This module will discuss one of the congruence postulates that

will set conditions for two triangle congruence.

What’s More

𝐴𝐵 1. ̅̅̅̅

𝐶𝐵 ≅ ̅̅̅̅

𝐵𝐷 , ̅̅̅̅

𝐵𝐷 ≅ ̅̅̅̅

𝐴𝐷 , ̅̅̅̅

𝐶𝐷 ≅ ̅̅̅̅

CDB∠ ≅ ADB∠ CBD,≅∠ ABD∠ C,∠ ≅ A ∠

∆MGA) ≅ ABS∆ 3. D (

What I have Learned

Two 1.

vertices 2.

corresponding sides 3.

corresponding angles 4.

What I can Do

A.

CONGRUENT ANGLES CONGRUENT SIDES

TL ̅̅̅̅

SC ≅ ̅̅̅

∠S ≅ ∠T

LE ̅̅̅̅

CI ≅ ̅

∠C ≅ ∠L

TE ̅̅̅̅

SI ≅ ̅

∠I ≅ ∠E

B.

AD 3. F∠ 2. GEO∆ 1. ̅̅̅̅

BE 4. ̅̅̅̅

∠PNM 5.

C.

A

R

M