8

Mathematics
Quarter 3 – Module 5:
Proves Two Triangles are
Congruent
Mathematics – Grade 8
Quarter 3 – Module 5: Proves Two Triangles are Congruent
First Edition,2020

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 8
Mathematics
Quarter 3 – Module 5:
Proves Two Triangles are
Congruent
Introductory Message

For the facilitator:

As a facilitator, you are expected to orient the learners on how to use
this module. You also need to keep track of the learners' progress while
allowing them to manage their own learning at home. Furthermore, you are
expected to encourage and assist the learners as they do the tasks included
in the module.

For the learner:

As a learner, you must learn to become responsible of your own
learning. Take time to read, understand, and perform the different activities
in the module.
As you go through the different activities of this module be reminded of
the following:
1. Use the module with care. Do not put unnecessary mark/s on any part
of the module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer Let Us Try before moving on to the other
activities.
3. Read the instructions carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your
answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are done.
If you encounter any difficulty in answering the tasks in this module,
do not hesitate to consult your teacher or facilitator. Always bear in mind that
you are not alone. We hope that through this material, you will experience
meaningful learning and gain deep understanding of the relevant
competencies. You can do it!

ii
 Let Us Learn

This module was designed and written with you in mind. It is here to help
you master the nature of Geometry. 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 them can be changed to
correspond with the textbook you are now using.

The module is composed of one lesson:

Lesson 1 – Proves two triangles congruent (M8GE-IIIg-1)

After going through this module, you are expected to:

1. define triangle congruent;

2. prove two triangles are congruent; and
3. relate triangle congruence in real- life situation.

1
 Let Us Try

Corresponding congruent parts are marked. Write their congruence.

O
Example:
U
E O
L V

C T K
t E
Figuret A t Figure B
#### ####, 𝐶𝑈
𝐶𝑇 ≅ 𝐸𝑂 #### ≅ ####
𝐸𝐾, #### ####
𝑇𝑈 ≅ 𝑂𝐾 #### ####, ####
𝐿𝑂 ≅ 𝑉𝑂 ####, ∠𝐿𝐸𝑂 ≅ ∠𝑉𝐸𝑂
𝑂𝐸 ≅ 𝑂𝐸

1.

2.

3.

2
 Let Us Study

Two geometric figures are congruent if they are of the same shapes and
measurement. In this chapter, we are going to discover the different conditions that
guarantee the congruence of two triangles.

Definition
Two triangles are congruent if their vertices can be paired so that corresponding
sides are congruent and corresponding angles are congruent.

To Prove two triangles congruent

1. Mark the diagram with the given.

2. Mark any additional parts that are equal.
3. If possible, label angles with numbers. This will make the job of writing the
proof easier.
4. Decide what postulate or theorem of proving triangles congruent to use.
5. Prove using two-column proof.

Examples:

Let us examine the following examples.

Example 1
Given:
#### ≅ 𝐷𝐸
𝐴𝐵 ####

∠𝐵 ≅ ∠𝐸
####
𝐵𝐶 ≅ ####
𝐸𝐹

Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹

Proof:
Statements Reasons
#### ≅ 𝐷𝐸
1. 𝐴𝐵 #### 1. Given
2. ∠𝐵 ≅ ∠𝐸 2. Given
#### ≅ 𝐸𝐹
3. 𝐵𝐶 #### 3. Given
4. ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 4. SAS Postulate

3
Example 2

Given:
#### ≅ 𝐴𝐷
𝐴𝐵 ####

####
𝐵𝐶 ≅ ####
𝐷𝐶

Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐷𝐶

Proof:

Statements Reasons
#### ≅ 𝐴𝐷
1. 𝐴𝐵 #### 1. Given
#### ≅ 𝐷𝐶
2. 𝐵𝐶 #### 2. Given
#### ≅ 𝐴𝐶
3. 𝐴𝐶 #### 3. Reflexive Property of Congruence

4. ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 4. SSS Postulate

C
Example 3

In ∆𝐴𝐶𝐵, let D be a point on 𝐴𝐵#### such that 𝐶𝐷

#### bisect
#### ####
∠ ACB. If 𝐴𝐶 ≅ 𝐵𝐶 , prove that ∆𝐴𝐶𝐷 ≅ ∆𝐵𝐶𝐷

A D B

Proof:

Statements Reasons
####
1. 𝐴𝐶 ≅ 𝐵𝐷#### 1. Given
#### 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐴𝐶𝐵
2. 𝐶𝐷 2. Given
3. ∠𝐴𝐶𝐵 ≅ ∠𝐵𝐶𝐷 3. Definition of angle bisector
#### ≅ 𝐶𝐷
4. 𝐶𝐷 #### 4. Reflexive property of congruence
5. ∆𝐴𝐶𝐷 ≅ ∆𝐵𝐶𝐷 5. SAS Congruence postulate

Example 4

Given:
####
𝐴𝐶 ≅ ####
𝐵𝐷
####
𝐴𝐶 ∥ ####
𝐵𝐷

Prove:

∆𝐴𝐶𝐸 ≅ ∆𝐵𝐷𝐸

4
Proof:

Statements Reasons
#### ≅ 𝐵𝐷
1. 𝐴𝐶 ####, #### ####
𝐴𝐶 ∥ 𝐵𝐷 1. Given
2. ∠𝐶 ≅ ∠𝐷 2. Parallel – Alternate Interior Angle
Postulate
3. ∠𝐴𝐸𝐶 ≅ ∠𝐵𝐸𝐷 3. Vertical Angles are congruent
4. ∆𝐴𝐶𝐸 ≅ ∆𝐵𝐷𝐸 4. SAA Congruence Theorem

Let Us Practice

Activity: Check your understanding

Complete each proof

1.
Given:
####
𝑅𝑌 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐴𝑅𝑀
#### #####
𝑅𝐴 ≅ 𝑅𝑀
Prove: ∆𝐴𝑌𝑅 ≅ ∆𝑀𝑌𝑅
Proof:

Statements Reasons
1. #### #####
𝑅𝐴 ≅ 𝑅𝑀 1.
2. ∠1 ≅ ∠2 #### 𝑏𝑖𝑠𝑒𝑐𝑡𝑠∠𝐴𝑅𝑀
2. 𝑅𝑌
3. #### ≅ #####
𝑅𝑌 𝑅𝑌 3.
4. ∆𝐴𝑌𝑅 ≅ ∆𝑀𝑌𝑅 4.

2.
Given:
∠ J and ∠ H are right angles

∠ 2 ≅ ∠ 4
Prove: ∆𝑂𝐽𝑁 ≅ ∆𝑁𝐻𝑂
Proof:

Statements Reasons
1. ∠ J and ∠ H are right angles 1.
2. 2. Given
3 ∠ J and ∠ H 3.
#### ≅ #####
4. 𝑁𝑂 𝑁𝑂 4.
5. ∆𝑂𝐽𝑁 ≅ ∆𝑁𝐻𝑂 5.

5
 #### such that 𝐶𝐷
3. In ∆ABC, let D be a point on 𝐴𝐵 #### ⊥ 𝐴𝐵
#### and 𝐴𝐷
#### #####
𝐵𝐷.

Prove that ∆ADC ∆ABC.

Proof:

STATEMENTS REASONS
#### ≅ #####
1. 𝐴𝐷 𝐵𝐷 1.

2. 2. Given

3.Definition of perpendicular line

3. ∠ ADC ≅ ∠ ________
segments
I ####
4. ####
𝐶𝐷 ≅ 𝐶𝐷 4.

5. ∆ADC ∆BDC 5.

Let Us Practice More

P
Write a two-column proof of the following.

1. Given: N is the midpoint of MO.

PN ^ MO
Prove: ∆ MPN ∆ONP
M N 0
Proof :
STATEMENTS REASONS

1. N is the midpoint of MO. PN ^ MO 1.

2. MN @ NO 2.

3. ∠ PNM 𝑎𝑛𝑑 ∠ PNO are rigth angles 3.

4. ÐPNM @ Ð 4. All right angles are congruent

5. PN @ PN 5.

6. ∆ MPN ∆ONP 6.

6
2. In the accompanying figure, CE @ CB
and ÐE @ ÐB.
Prove that D DCE @ D ACB
E
D
C
A B

Statements Reasons
1. CE @ CB 1.
2.ÐE @ ÐB 2.
3. ÐDCE @ ÐACB 3.
4. D DCE @ D ACB 4.

Let Us Remember
This lesson was about proving two triangles. Apply the postulates and
theorems on triangle congruence to prove statements on congruence, including right
triangles.

Let Us Assess

Multiple Choice. Choose the letter of the best answer. Write your answer on your
answer sheet.
#### ≅ 𝑋𝑌
1. Which of the following properties justifies the statement “𝑋𝑌 #### ”?
A. reflexive B. symmetric C. transitive D. equivalence

2. 𝛥𝐷𝐵𝐴 ↔ 𝛥𝐸𝐹𝐺. Which angle corresponds to ∠𝐸𝐺𝐹?

A. ∠𝐵𝐶𝐷 B.∠𝐶𝐷𝐵 C. ∠ BAD D.∠𝐶𝐷𝐵

3. In 𝛥𝐿𝑀𝑁, which is the included side of ∠𝐿 and ∠𝑀?

A. LM #####
B.𝑀𝑁 ####
C.𝐿𝑁 ####
D.𝑁𝐿

4. In 𝛥𝐶𝐷𝐸, which is the included angle of sides #### ####?

𝐶𝐸 and 𝐶𝐷
A. ÐC B.∠𝐷 C.∠𝐸 D.∠𝐶𝐷𝐸

5. If ∆𝑆𝑈𝑀 = ∆𝑃𝑅𝑂, which angle is congruent to ∠𝑀?

A. ∠𝑆 B. ∠𝑃 C. ∠𝑅 D. ∠ O

6. Which pair of triangles shows congruency by the SAS Postulate?

7
 Figure A Figure B Figure C Figure D
A. Figure A B. Figure B C. Figure C D. Figures C and D
7. Which postulate can be used to prove that the triangles are congruent?

A. SAS Postulate B. ASA Postulate C. SSS Postulate D.SAA Postulate

8. ∆𝐴𝐵𝐶 is congruent to ∆𝐷𝐸𝐹 by ASA Postulate. What other congruence are shown
by the triangles using CPCTC?
A. ∠𝐵 ≅ ∠𝐸 B. ∠𝐵 ≅ ∠𝐸
####
𝐵𝐶 ≅ ####
𝐸𝐹 #### ≅ ####
𝐴𝐵 𝐸𝐷
∠𝐶 ≅ ∠𝐹 ∠𝐶 ≅ ∠𝐹

C. D.
####
𝐵𝐶 ≅ ####
𝐸𝐹 #### ≅ ####
𝐴𝐵 𝐷𝐸
∠𝐴 ≅ ∠𝐷 #### ≅ 𝐷𝐹
𝐴𝐶 ####

9. Tina had proven that ∆𝐹𝐽𝐻 ≅ ∆𝐻𝐺𝐹 by SSS Postulate. She said that ∠𝐻 ≅ ∠𝐹.
What lets Tina claim that the two angles are also congruent?

A. by SSS Postulate B. by CPCTC

C. by triangle congruence D. Isosceles Triangle Theorem

10. Lloyd knows that AB = XY and AC = XZ. What other information must he know
to prove ΔABC ≅ ΔXYZ by SAS postulate?
A. ∠B ≅ ∠Y B. ∠C ≅ ∠Z
C. ∠A ≅ ∠X D. not enough information
11. ΔTIN ≅ ΔCAN, then ΔNAC is congruent to ____.
A. ΔITN B. ΔNIT C. ΔTNI D. ΔINT
12. ΔABC ≅ ΔDEF, which segment is congruent to AB ?
A. BC B. AC C. DE D. EB
13.What do you mean by CPCTC?

A. Corresponding parts of congruent triangles are congruent

B. Classifying parts of congruent triangles are congruent

8
 C. Corresponding parts of corresponding triangles are congruent
D. none of the above

14. If D YXZ @ D MTS, which of the following is true?

A. XY @ TS B. ÐXYZ @ ÐSMT C. XZ @ TM D. ÐZXY @ ÐMST

15. If #### ####, ∠𝐻 ≅ ∠𝑁 and ∠𝑂 ≅ ∠𝑅, which triangle – congruence postulate

𝐻𝑂 ≅ 𝑁𝑅
guarantees that ∆𝐻𝑂𝑃 ≅ ∆𝑁𝑅𝑇?
A. SAS Postulate B. ASA Postulate C. SSS Postulate D.SAA Postulate

Let Us Enhance

Performance task

Take a picture of object in your house or surrounding where you can see triangles.
For each picture, identify congruent triangles. Justify why these triangles are
congruent. Make a portfolio of this task.

RUBRIC

CRITERIA OUTSTANDING SATISFACTORY DEVELOPING BEGINNING RATING

10 8 6 3

The The The The

explanation explanation explanation explanatio
is clear, is clear and in n is
Mathematical exhaustive, coherent. It understanda incomplete
Reasoning or thorough covers the ble but not and
and important logical inconsisten
coherent. It concepts. t
includes
interesting
facts and
principles

The design The design is The design The design

is presentable makes use does not
Creativity comprehensi and makes of geometric use
ve and use of the representati geometric
display concepts of ons but not representat
aesthetic the presentable ion and not
aspect of the geometric presentabl
mathematica representati e
l concepts on
learned

9
 Let Us Reflect

Like, Heart and Sad Reacts

On the Like React, write three things that you have learned about the lesson.
On the Heart React, write two real-life situations where you can apply the concept of
triangle congruence
On the Sad React, write one question that you want to ask about the topic.

10
 11
Let us Assess
1. A
2. C
3.A
4.A
5.D
6.C
7.B
8.A
9.B
10C
11.B
12.C
13.A
14.B
15.B
Let us practice more Let Us Practice
I. Let Us Try
1.) 1. Given 1. AB @DE
3. Reflexive
BC @ EF
property of
1. Given congruence ÐB @ ÐE
4. SAS congruence SAS Postulate
2. Definition of postulate
midpoint 2. AB @ DE
II. AC @ DF
3. Definition of 1. Given
perpendicular lines BC@ EF
2. Ð2 @ Ð4
3. Definition of SSS Postulate
4.ÐPNO right angle 3. ÐC @ ÐF
5. Reflexive 4. Reflexive CB @ EF
property of
property of ÐB @ÐF
congruence
congruence 5. SAA congruence ASA Postulate
theorem
6. SAS postulate
2.) III.
1. Given
1. Given 2. CD ^ AB
3. ÐBDC
2. Given 4. Reflexive
3. Vertical Angle property of
congruence
theorem
5. SAS congruence
4. ASA postulate postulate
Answer Key
 References

Emmanuel P. Abuso, et.al., Mathematics Grade 8 Learner`s

Module,Department of Education- Bureau of learning Resources 2013,
358-361

Evangeline P. Bautista, Ph.D. et.al., Experiencing Mathematics Geometry,

Vibal Publishing House,2006, 118-120

Simon L. Chua,D.T. et.al., Soaring 21st Century Mathematics Exploring

Geometry, Phoenix Publishing House,2004,243-248
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