Mathematics 8 Quarter 3 Module 5: Solving Corresponding Parts of Congruent Triangles

Mathematics

Quarter 3 – Module 5 Solving Corresponding Parts of Congruent Triangles

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[illustration: students and a teacher studying together, with a globe, a lamp, and books]

Mathematics – Grade 8 Alternative Delivery Mode Quarter 3 – Module 5 Solving Corresponding Parts of Congruent Triangles First Edition, 2020

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

Development Team of the Module

Writers: Nilbeth S. Merano, Dexter M. Abellanosa Language Editor: Joel B. Asonto Content Editors: Merjorie G. Dalagan, Rosalita A. Bastasa, Melissa R. Cabarles Lay-out Editor: Ivin Mae N. Ampons Reviewers: Rhea J. Yparraguirre, Manuel L. Limjoco, Jr., Cris Gerom C. Arguilles, Melba G. Lumangcas, Glorina M. Arreza, Joysie P. Sabejon Illustrators: Nilbeth S. Merano, Dexter M. Abellanosa, Prince Conel R. Mamolang, Kirby D. Caballero, Marieto Cleben V. Lozada Layout Artists: Nilbeth S. Merano, Dexter M. Abellanosa, Joliver R. Reposposa, Leonil Rechie P. Cahanap Management Team: Francis Cesar B. Bringas, Isidro M. Biol, Jr., Maripaz F. Magno, Josephine Chonie M. Obseñares, Josita B. Carmen, Celsa A. Casa, Regina Euann A. Puerto, Bryan L. Arreo, Elnie Anthony P. Barcenas, Leopardo P. Cortes, Jr., Claire Ann P. Gonzaga

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Mathematics

Quarter 3 – Module 5

Solving Corresponding Parts of Congruent Triangles

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 learn the skills in solving problems about congruent triangles. You are provided with varied activities to reinforce your knowledge and skills and to deepen your understanding of the lesson. The scope of this module enables you to use it in different learning situations. The lesson is arranged to follow the standard sequence of the course. But the order in which you read them can be rearranged to correspond with the textbook you are using.

This module contains a single lesson:

Lesson 1- Solving Corresponding Parts of Congruent Triangles

After going through this module, you are expected to:

1. identify corresponding parts of congruent triangles;
2. name congruent triangles;
3. find the measure of corresponding parts of congruent triangles; and
4. relate triangle congruence to real life situations.

What I Know

PRE-ASSESSMENT

Directions: Choose the letter of the correct answer and write it on a separate sheet of paper.

1. Given that ∆ABC ≅ ∆DEF, what angle corresponds to ∠A?

   • A. ∠B
   • B. ∠C
   • C. ∠D
   • D. ∠F

2. Given that ∆MPC ≅ ∆STW, what segment corresponds to M̅C̅?

   • A. C̅S̅
   • B. M̅S̅
   • C. M̅T̅
   • D. S̅W̅

3. If ∆ABC ≅ ∆XYZ, which of the following is NOT true?

   • A. ∠A ≅ ∠X
   • B. ∠C ≅ ∠Z
   • C. A̅B̅ ≅ Y̅X̅
   • D. B̅C̅ ≅ Y̅Z̅

4. If ∆GHI ≅ ∆JKL, which of the following are corresponding segments?
   I. G̅I̅ and J̅L̅ II. K̅J̅ and H̅G̅ III. G̅H̅ and K̅J̅

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

5. Which of the following is a congruence statement in Figure 1?

   • A. ∆LNM ≅ ∆PQR
   • B. ∆MLN ≅ ∆QRP
   • C. ∆NLM ≅ ∆PRQ
   • D. ∆NLM ≅ ∆RPQ

[chart: diagram; Figure 1] Labels: M, L, Q, N, P, R. [transcriber’s note: the figure shows a quadrilateral with a diagonal segment from N to Q, right-angle markers, and angle arcs. No values are printed.]

6. If ∠L ↔ ∠J, ∠B ↔ ∠R and ∠C ↔ ∠S, then ∆LBC ≅ ____.

   • A. ∆BJC
   • B. ∆CSB
   • C. ∆JRS
   • D. ∆LRS

7. If M̅N̅ ≅ P̅Q̅, M̅O̅ ≅ P̅R̅, N̅O̅ ≅ Q̅R̅, then ∆MON ≅ ____.

   • A. ∆PRQ
   • B. ∆PQR
   • C. ∆RPQ
   • D. ∆RQP

8. If ∆LRT ≅ ∆CAN, which of the following congruent statements is/are true?
   I. ∆LTR ≅ ∆CNA II. ∆TRL ≅ ∆NCA III. ∆RTL ≅ ∆ANC

   • A. I and II
   • B. I and III
   • C. II and III
   • D. I, II, and III

9. If ∆TUV ≅ ∆WXY, T̅U̅ is 30 cm, find the length of WX.

   • A. 10 cm
   • B. 20 cm
   • C. 30 cm
   • D. 40 cm

10. Given that ∆MOB ≅ ∆CAN, ∠M = 50° and ∠B = 60°, find ∠A.

    • A. 50°
    • B. 60°
    • C. 70°
    • D. 110°

11. ∆BEL is congruent to ∆DON. Find the value of x in Figure 2.

    • A. 5
    • B. 6
    • C. 14
    • D. 30

[chart: diagram; Figure 2] Labels: B, E, L, D, N, O. Angle values: 30° at B; (5x)° at E; (2x+2)° at D.

12. Beatriz makes a ribbon pattern for her art class. Figure 3 shows the pattern with marks indicating which lengths are equal. If CE = 7 cm and EG = 4 cm, find EF.
    • A. 3 cm
    • B. 4 cm
    • C. 7 cm
    • D. 11 cm

[chart: diagram; Figure 3] Labels: C, E, F, D, G. Equal-length marks are shown on the segments as printed; no values are printed other than those in the question.

13. Riza and Leone are cutting congruent triangles out of paper to be used as decorations for Ebenezer’s birthday. Figure 4 shows some of the measurements of the triangles. Find the value of x.
    • A. 4
    • B. 5
    • C. 6
    • D. 7

[chart: diagram; Figure 4] Labels along the top line: 6 in.; (3x-6) in.

14. Figure 5 shows that ∆MET ≅ ∆RST. Can you say that ∠MET = 33° and ∠TRS = 48°?
    • A. Yes, ∠MET = 33° and ∠TRS = 48°.
    • B. No, ∠MET = 48° and ∠TRS = 60°.
    • C. No, ∠MET = 60° and ∠TRS = 33°.
    • D. It is impossible to determine their measures.

[chart: diagram; Figure 5] Labels: M, R, T, E, S. Angle labels: (4y+3)° at T (left), (5y-21)° at T (right), (2x)° at R [?].

15. Mr. Baguhin owns a residential lot in the province. Figure 6 shows the measurements of the lot. Which of the following is true?
    • A. EV = 13 m, RV = 14 m, SV = 12 m
    • B. EV = 14 m, RV = 15 m, SV = 13 m
    • C. EV = 20 m, RV = 17 m, SV = 14 m
    • D. EV = 16 m, RV = 18 m, SV = 15 m

[chart: diagram; Figure 6] Labels: S, V, E, R. Side labels: (5y+4) m [?] on SE; (5y+2) m [?] on VR; (4y+2) m on ER; (7y-1) m on diagonal EV. [transcriber’s note: the side labels on the left and right edges are rotated and only partly legible; the readings marked [?] are best readings.]

Lesson 1: Solving Corresponding Parts of Congruent Triangles

Have you ever been amazed looking at those towering buildings in cities and in movies? Have you ever wondered how such skyscrapers withstand strong winds and heavy rains? Think about bridges this time, why do you think they appear so calm while enduring the rapid flow of water below and the tons of load above?

Considered as the most stable of all geometric figures, triangles. Triangles play a vital role in a structure’s stability. Congruent triangles, in particular, are used in architectural and engineering design to make structures strong and stable.

[illustration: a tall skyscraper and a truss bridge]

STABILITY. Congruent triangles play an important role in architecture and engineering. Trusses, which are made of triangulated system of materials, support buildings and bridges so they can withstand the tests of [illegible]

In this module, you will not just explore the world of congruent triangles but even learn how to solve their corresponding parts.

What’s In

Activity: Vice Versa

You have learned in the previous module the different conditions that make triangles congruent. You also learned that when two triangles are congruent, there are 3 corresponding angles and 3 corresponding sides, which are also congruent. This time, your task is to complete the table below by providing the missing triangle congruence statement or the missing congruent corresponding parts. The first one has been done for you.

Congruence Statement	Corresponding Angles				Corresponding Sides
∆CAT ≅ ∆NEL	∠C ≅ ∠N	∠A ≅ ∠E	∠T ≅ ∠L	CA̅ ≅ NE̅	AT̅ ≅ EL̅	CT̅ ≅ NL̅
∆DSL ≅ ∆FMR
	∠L ≅ ∠B	∠O ≅ ∠A	∠W ≅ ∠T
				KR̅ ≅ PC̅	RB̅ ≅ CM̅	KB̅ ≅ PM̅

Questions:

1. Did you find difficulty in answering the activity above?
2. Were you able to determine the corresponding angles and corresponding sides given the triangle congruence statement?
3. How did you write the triangle congruence statement given the corresponding angles and corresponding sides?

What’s New

Let us now begin by exploring the world of congruent triangles and the important role they play in construction. Read and do the tasks that follow.

The figure at the right is a truss. It is a triangulated system where straight structural elements such as beams, bars or rods are interconnected. It is a strong frame that supports a building or a bridge.

[illustration: truss, a triangulated frame of straight structural members]

Suppose we are going to assign points on the truss as shown below.

[illustration: truss with points labeled L, I, E, N, Z, M, and R]

1. List down all possible pairs of triangles that appear to be congruent. What was your basis in choosing those pairs of triangles?

   
   

2. Consider the two triangles ∆NIM and ∆ZER to be congruent. We have the following 6 pairs of corresponding parts:

   
   

   ∠N ≅ ∠Z ⟶ If ∠N = 28°, then ∠Z = _______.
   ∠I ≅ ∠E ⟶ If ∠I = 92°, then ∠E = _______.
   ∠M ≅ ∠R ⟶ If ∠M = 60°, then ∠R = _______.
   N̅I̅ ≅ Z̅E̅ ⟶ If NI = 4 ft, then ZE = _______.
   I̅M̅ ≅ E̅R̅ ⟶ If IM = 3 ft, then ER = _______.
   M̅N̅ ≅ R̅Z̅ ⟶ If MN = 5 ft, then RZ = _______.

   
   

What is It

Two triangles are congruent if all of their parts coincide. That is, for the two triangles to be congruent, they must have the same shape and the same size.

Consider the triangles at the right. Suppose ∆CAB is made to coincide with ∆OFX such that the vertices of ∆CAB fit exactly over the vertices of ∆OFX, there exists a correspondence between vertices such that:

C ↔ O A ↔ F B ↔ X

[illustration: triangle CAB and triangle OFX, with matching tick marks and arcs on corresponding sides and angles]

C ↔ O is read as “vertex C corresponds to vertex O,” and so forth.

Doing the same process, if ∆CAB is made to coincide again with ∆OFX such that the sides of ∆CAB fit exactly over the sides of ∆OFX, there exists a correspondence between sides such that:

C̅A̅ ↔ O̅F̅ A̅B̅ ↔ F̅X̅ C̅B̅ ↔ O̅X̅

CA̅ ↔ OF̅ is read as “side CA corresponds to side OF,” and so forth.

The angles and the sides will match up this way:

Corresponding Angles	Corresponding Sides
∠C ↔ ∠O	C̅A̅ ↔ O̅F̅
∠A ↔ ∠F	A̅B̅ ↔ F̅X̅
∠B ↔ ∠X	C̅B̅ ↔ O̅X̅

Definition

Given a correspondence CAB ↔ OFX between vertices of two triangles. If every pair of corresponding sides are congruent, and every pair of corresponding angles are congruent, then the correspondence CAB ↔ OFX is called a congruence between the two triangles.

When we write ∆CAB ≅ ∆OFX, we mean that the correspondence CAB ↔ OFX is a congruence. The single expression ∆CAB ≅ ∆OFX also tells us the following:

Corresponding Angles	Corresponding Sides
∠C ≅ ∠O or m∠C = m∠O	C̅A̅ ≅ O̅F̅ or CA = OF
∠A ≅ ∠F or m∠A = m∠F	A̅B̅ ≅ F̅X̅ or AB = FX
∠B ≅ ∠X or m∠B = m∠X	C̅B̅ ≅ O̅X̅ or CB = OX

In general, two triangles are congruent if and only if their vertices can be matched up so that the corresponding parts (angles and sides) of the triangles are congruent.

When two triangles are congruent, we put marks to indicate congruences between angles and sides. Angles and sides with the same marks are congruent. Thus, they have the same measure.

In the figure at the right, both ∠C and ∠O are marked with one arc, hence, the two are congruent angles. On the other hand, both C̅B̅ and O̅X̅ have two small slash marks that tell us they are congruent.

[illustration: triangle CAB and triangle XFO, with one arc marked on ∠C and ∠O, and two slash marks on CB̅ and OX̅]

In addition, we can write congruence statements in several ways. Take for instance the congruence statement ∆CAB ≅ ∆OFX, we can write this as: ∆CBA ≅ ∆OXF, ∆ACB ≅ ∆FOX, ∆ABC ≅ ∆FXO, ∆BCA ≅ ∆XOF and ∆BAC ≅ ∆XFO. When referring to congruent triangles, name their corresponding vertices in the same order.

Remember

• Two triangles are congruent if and only if their vertices can be paired so that corresponding sides and corresponding angles are congruent.
• Corresponding parts of congruent triangles are congruent (CPCTC).

Let’s take a look at the following examples. Remember that the concepts discussed above will help you solve problems involving congruent triangles.

EXAMPLE 1

a.) Given that ∆AIM ≅ ∆SET, list down the corresponding parts.

Answer:

Corresponding Angles	Corresponding Sides
∠A ↔ ∠S	A̅I̅ ↔ S̅E̅
∠I ↔ ∠E	I̅M̅ ↔ E̅T̅
∠M ↔ ∠T	A̅M̅ ↔ S̅T̅

b.) Name the congruent triangles with the following congruent corresponding parts:

∠M ≅ ∠C, ∠A ≅ ∠P, ∠T ≅ ∠U and M̅A̅ ≅ C̅P̅, A̅T̅ ≅ P̅U̅, M̅T̅ ≅ C̅U̅

Answer: ∆MAT and ∆CPU

EXAMPLE 2

a.) Given: ∆IEF ≅ ∆HGF Find: ∠H

Answer: ∠H = 40°

[illustration: triangles IEF and HGF sharing vertex F, with points E, G, H, I labeled and 40° marked at I]

Reason: ∠H ≅ ∠I ,

b.) Given: ∆ABC ≅ ∆ADC Find: y

[illustration: triangles ABC and ADC sharing side AC, with DA labeled 3y cm and AB labeled 18 cm]

Answer: y = 6

Solution: Since A̅B̅ ≅ A̅D̅,

3y = 18 Definition of Congruent Segments

(1/3) 3y = (1/3) 18 Multiply (1/3); Multiplication Property of Equality

y = 6 Result

EXAMPLE 3

Given that ∆MYX ≅ ∆JAN, ∠M = 40° and ∠A = 70°, find ∠X.

Solution

Since ∆MYX ≅ ∆JAN, the corresponding parts, particularly, the angles are congruent. By CPCTC, we can say that ∠M ≅ ∠J, ∠Y ≅ ∠A and ∠X ≅ ∠N. To find ∠X, we need to focus on ∆MYX. We know that ∠M = 40° and since ∠Y ≅ ∠A, we can say that ∠Y = 70°.

∠M + ∠Y + ∠X = 180° Angle Sum Theorem 40° + 70° + ∠X = 180° Substitute ∠M = 40° and ∠Y = 70° 110° + ∠X = 180° Simplify 110° – 110° + ∠X = 180° – 110° Add (-110°); Addition Property of Equality ∠X = 70°. Result

Example 4

∆JOY is congruent to ∆FUL. Find the value of n and ∠L.

[illustration: triangle JOY with 60° at J and 70° at O; triangle FUL with (2n+10)° at U]

Solution

We know that when ∆JOY ≅ ∆FUL, the corresponding angles ∠U and ∠O are congruent. Thus, they have the same measures.

2n+10 = 70 Substitute m∠U = 2n+10 and m∠O = 70 2n+10-10 = 70-10 Add (-10); Addition Property of Equality (1/2) 2n = (1/2) 60 Multiply (1/2); Multiplication Property of Equality n = 30. Result

To find ∠L, we note that it corresponds to ∠Y. To come up with a total measure of 180° in ∆JOY, ∠Y must be 50°. Since ∠Y ≅ ∠L, we can conclude that ∠L = 50°.

Example 5

The figure at the right shows that ∆BOE ≅ ∆TXE. Solve for x and find ∠EOB.

[illustration: triangles BOE and TXE meeting at E, with (4x-2)° and (3x+23)° marked at E, (2x-5)° marked at T, and an angle marked at B]

Solution

By CPCTC, ∠OEB ≅ ∠XET. So, we say that their measures are equal.

4x-2 = 3x+23 Definition of Congruent Angles 4x-3x-2 = 3x-3x+23 Add (-3x); Addition Property of Equality x-2 = 23 Simplify x-2+2 = 23+2 Add 2; Addition Property of Equality x = 25 Result

We cannot right away solve for ∠EOB since there is no expression assigned to it. Let us first get the measures of the other two angles in ∆BOE.

For ∠BEO:

∠BEO = (4x-2)° Given ∠BEO = [ 4(25)-2 ]° Substitute x by 25 ∠BEO = (100-2)° Simplify ∠BEO = 98° Result

For ∠OBE:

∠OBE = (2x-5)° ∠OBE ≅ ∠XTE; Definition of Congruent Angles ∠OBE = [2(25)-5]° Substitute x by 25 ∠OBE = (50-5)° Simplify ∠OBE = 45° Result

We already solved for the other two angles in ∆BOE, we can now proceed to finding the measure of ∠EOB. Don’t forget that the total measure of all the angles in a triangle is 180°.

∠BEO + ∠OBE + ∠EOB = 180° Angle Sum Theorem 98° + 45° + ∠EOB = 180° Substitute ∠BEO=98°, ∠OBE=45° 143° + ∠EOB = 180° Simplify 143° – 143° + ∠EOB = 180° – 143° Add (-143°); Addition Property of Equality ∠EOB = 37° Result

Hence, x = 25 and ∠EOB = 37°.

What’s More

Activity 1. Giving It All

Directions: Give what is being asked in each problem.

1. Given that ∆FUN ≅ ∆DAY. List down all corresponding parts.

[illustration: triangle FUN and triangle DAY with matching tick marks and arcs]

2. Name the congruent triangles having the following corresponding parts:

∠S ≅ ∠C, ∠I ≅ ∠O, ∠T ≅ ∠M and S̅I̅ ≅ C̅O̅, I̅T̅ ≅ O̅M̅, S̅T̅ ≅ C̅M̅

Questions:

1. How did you find the activity?
2. Is it difficult to identify the corresponding parts of congruent triangles?
3. How did you name the congruent triangles given the corresponding parts?

Activity 2. Figuring It Out

Directions: Given the following figures, find the value of the indicated variable.

1. ∆ABC ≅ ∆ADC. Find h.

[illustration: triangle ABD with altitude AC to base DB, a right-angle mark at C, and sides labeled (5h-7) cm and (3h+21) cm]

2. ∆WXY ≅ ∆VYX. Find m.

[illustration: quadrilateral with diagonals crossing, labeled X, Y, W, V, with segments 4 ft, (2m) ft, 18 ft, and (7m+13) ft]

Questions:

1. Did you find difficulty in performing the task above? If yes, share your thoughts.
2. Were you able to get the value of the variables?
   If yes, explain how you did it. If no, cite the reason/s.

Activity 3. Thinking It Over

Directions: Determine whether each statement is true or false. Check the box of your answer. If true, support it. If false, give a counterexample. Write your answer on a separate sheet of paper.

a. If two triangles are congruent, their perimeters are equal.

True ☐ False ☐ ______________________________________

b. If two triangles have the same perimeter, they are congruent.

True ☐ False ☐ ______________________________________

What I Have Learned

BLANK SPACES BETWEEN US

Directions: Complete the flowchart below on a separate sheet of paper to summarize what you have learned about solving corresponding parts of congruent triangles.

[chart: flowchart; SOLVING CORRESPONDING PARTS OF CONGRUENT TRIANGLES]

SOLVING CORRESPONDING PARTS OF CONGRUENT TRIANGLES

CPCTC stands for

C_______ P_______ of C_______ T _______ are C_______.

Diagram labels:

• Left triangle (vertices C, A, B): 4 cm; (5x+10)° at A; 6 cm; 8 cm
• Right triangle (vertices F, D, E): (2a) cm; 55° at D; (2b) cm; (4c) cm

What angle is congruent to ∠A?	What side is congruent to $\overline{AC}$?	What side is congruent to $\overline{AB}$?	What side is congruent to $\overline{BC}$?
What is the value of x?	What is the value of a?	What is the value of b?	What is the value of c?
What is the measure of ∠A?	What is the length of $\overline{AC}$?	What is the length of $\overline{AB}$?	What is the length of $\overline{BC}$?

What I Can Do

MEASURING A RIVER

How Napoleon’s Engineer Found a River’s Width without Instruments

Did you experience being asked to find the length or the width of an object without directly measuring it? For instance, you finding the width of the river without actually crossing it. Do you think it is possible?

Find it out by reading the story below and do the task that follows.

[illustration: riverbank scene with a man standing near a house on the left; points labeled B, E, T, and S]

Legend has it that the great emperor, Napoleon, while standing on the bank of a wide river, ordered one of his officials to determine its width so he could attack the enemy with artillery on the other side.

The official answered that he needed to get his instruments to carry out the order. But the emperor insisted that he required an immediate answer.

The official quickly devised a plan. Standing straight on the riverbank, he lowered the visor of his cap until the farthest thing he could see was the edge of the opposite bank. While maintaining the angle for which he bent his head, he slowly turned and noted the spot that was in line with his eye and the tip of his visor. And there he had it! He was able to give the emperor the width of the river.

Task: It’s Your Turn!

If you were the official faced with the challenge of determining the width of the river without actually measuring it, would you do the same? Why or why not? Support your answer by stating your plan in solving the challenge.

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Assessment

POST-ASSESSMENT

Directions: Read the questions carefully and choose the letter of the correct answer. Write your answer on a separate sheet of paper.

1. Given that ∆ALS ≅ ∆BKD, what segment corresponds to $\overline{AS}$?

   • A. $\overline{AK}$
   • B. $\overline{BD}$
   • C. $\overline{LB}$
   • D. $\overline{SK}$

2. Given that ∆HIJ ≅ ∆LMN, what angle corresponds to ∠J?

   • A. ∠H
   • B. ∠I
   • C. ∠M
   • D. ∠N

3. If ∆SIN ≅ ∆COS, which of the following is NOT true?

   • A. ∠I ≅ ∠O
   • B. ∠S ≅ ∠C
   • C. $\overline{IN}$ ≅ $\overline{OC}$
   • D. $\overline{NI}$ ≅ $\overline{SO}$

4. If ∆LUV ≅ ∆YAH, then ∆HAY is congruent to ____.

   • A. ∆AYH
   • B. ∆LVU
   • C. ∆UVL
   • D. ∆VUL

5. If ∠E ↔ ∠N, ∠V ↔ ∠S and ∠R ↔ ∠M, then ∆EVR ≅ ____.

   • A. ∆NSM
   • B. ∆MNS
   • C. ∆NSR
   • D. ∆SRV

6. Which of the following is a congruence statement in Figure 1?

   
   
   
   
   • A. ∆BVS ≅ ∆VLB
   
   
   • B. ∆ISV ≅ ∆BNL
   
   
   • C. ∆SVI ≅ ∆LBN
   
   
   • D. ∆VSI ≅ ∆NBL
   
   
   
   

   [illustration: Figure 1, a right-angled figure with vertices I, V, L, S, B, and N; angle marks shown]

   
   

   Figure 1

   
   

7. Given that ∆VAN ≅ ∆BUS, ∠V = 60° and ∠N = 70°, find ∠U.

   • A. 50°
   • B. 60°
   • C. 70°
   • D. 80°

8. If ∆BUN ≅ ∆DLE, BN is 3.15 cm, BU is 5.90 cm find the length of DE.

   • A. 2.75 cm
   • B. 3.15 cm
   • C. 5.90 cm
   • D. 9.05 cm

9. Live has two triangular vegetable gardens shown in Figure 2. He wants to fence them to protect his crops. It takes 7 m of fencing material for one garden. What is the total length of fencing material needed if the two gardens are of the same size and shape?

   [illustration: Figure 2, two triangles meeting at a point, with tick marks on their sides]

   • A. 7 m
   • B. 14 m
   • C. 21 m
   • D. 28 m

10. If ∆ACT ≅ ∆NIV, which of the following are corresponding segments?

    I. $\overline{TA}$ and $\overline{VN}$
    II. $\overline{CA}$ and $\overline{NI}$
    III. $\overline{TC}$ and $\overline{IV}$

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

11. If ∆LOC ≅ ∆KER, which of the following congruent statements is/are NOT true?

    
    

    I. ∆CLO ≅ ∆REK
    II. ∆COL ≅ ∆KRE
    III. ∆LCO ≅ ∆KER

    
    
    
    
    • C. I only
    
    
    • D. II only
    
    
    • C. I and III
    
    
    • D. I, II and III
    
    
    
    

    transcriber’s note: the printed answer choices for item 11 label two options “C.” and two options “D.”; they are transcribed as printed.

    
    

12. Figure 3 shows that ∆BEN is congruent to ∆IVR. Find the value of x.

    [illustration: Figure 3, two triangles with vertices E, N, B, I, V, and R; angle labels 41°, (5x)°, and (3x+2)°]

    • A. 5
    • B. 6
    • C. 13
    • D. 30

13. You are making party hats for your bestfriend’s birthday. The measurements are shown in Figure 4. Find the value of x.

    [illustration: Figure 4, two triangular party hats side by side; labels (2x-6) in. and 8 in.]

    • A. 3
    • B. 5
    • C. 7
    • D. 9

14. Cielo is tasked to complete the labels for an output in Math shown in Figure 5. She must see to it that each triangle should have a perimeter of 46 cm. What possible algebraic expressions should she choose to represent the dimensions of the given figure?

    [illustration: Figure 5, a quadrilateral with vertices L, M, K, and H, a diagonal LK labeled (2x+14) cm, and tick marks on its sides]

    • A. HL = (2x+6) cm, LM = (2x+14) cm, KM = (x+7) cm
    • B. HL = (4x+6) cm, LM = (3x+3) cm, KM = (2x+10) cm
    • C. HL = (5x+9) cm, LM = (x+14) cm, KM = (7x+1) cm
    • D. HL = (3x+9) cm, LM = (8x-3) cm, KM = (7x+1) cm

15. Figure 6 shows that ∆FUN ≅ ∆DUB. Do you agree that ∠FUN = 94° and ∠DBU = 42°?

    [illustration: Figure 6, two crossing triangles with vertices D, B, U, N, and F; angle labels (6y)°, (13y+3)°, and (10y+24)°]

    • A. Yes, ∠FUN = 94° and ∠DBU = 42°.
    • B. No, ∠FUN = 94° and ∠DBU = 44°.
    • C. No, ∠FUN = 90° and ∠DBU = 42°.
    • D. It is impossible to determine their measures.

Additional Activities

The following activity will help you enrich your knowledge on the concepts that you have just learned.

Directions: Create your own word problem in real life about solving corresponding parts of congruent triangles. Use the given objects as your bases in creating the word problem. Show the process in solving in a separate sheet of paper.

REAL LIFE APPLICATION	OWN WORD PROBLEM	SOLUTION PROCESS
UMBRELLA [photo: a smiling woman holding a patterned umbrella] Photo: Ebenezer S. Merano
BRIDGE [illustration: a truss bridge over water]
HOOD BRACES [illustration: a car with its hood open, held by braces]

Answer Key

transcriber’s note: the answer key tables on this page are printed upside down (rotated 180°) relative to the heading. The content is transcribed below in the order of the module’s sections, upright.

What I Know

1. C
2. D
3. C
4. B
5. A
6. C
7. A
8. B
9. C
10. C
11. C
12. C
13. A
14. A
15. C

What’s In

Congruence Statement	Corresponding Angles	Corresponding Sides
∆CAT ≅ ∆NEL	∠C ≅ ∠N ∠A ≅ ∠E ∠T ≅ ∠L	$\overline{CA}$ ≅ $\overline{NE}$ $\overline{AT}$ ≅ $\overline{EL}$ $\overline{CT}$ ≅ $\overline{NL}$
∆DSL ≅ ∆FMR	∠D ≅ ∠F ∠S ≅ ∠M ∠L ≅ ∠R	$\overline{DS}$ ≅ $\overline{FM}$ $\overline{SL}$ ≅ $\overline{MR}$ $\overline{DL}$ ≅ $\overline{FR}$
∆LOW ≅ ∆BEAT [?]	∠L ≅ ∠B ∠O ≅ ∠A ∠W ≅ ∠T	$\overline{LO}$ ≅ $\overline{BA}$ $\overline{OW}$ ≅ $\overline{AT}$ $\overline{LW}$ ≅ $\overline{BT}$
∆KRB ≅ ∆PCM	∠K ≅ ∠P ∠R ≅ ∠C ∠B ≅ ∠M	$\overline{KR}$ ≅ $\overline{PC}$ $\overline{RB}$ ≅ $\overline{CM}$ $\overline{KB}$ ≅ $\overline{PM}$

What’s New

1. varied answers
2. 28°
   92°
   60°
   4 ft
   3 ft
   5 ft

What’s More

Activity 1

1. ∠F ≅ ∠D
   ∠U ≅ ∠A
   ∠N ≅ ∠Y
   $\overline{FU}$ ≅ $\overline{DA}$
   $\overline{UN}$ ≅ $\overline{AY}$
   $\overline{FN}$ ≅ $\overline{DY}$

Activity 2

1. 14
2. 2

Activity 3

1. True (CPCTC)
2. False (varied counterexamples)

What I Have Learned

CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent.

[chart: flowchart; answers to the What I Have Learned activity]

• What angle is congruent to ∠A? ∠D
• What side is congruent to $\overline{AC}$? $\overline{DF}$
• What side is congruent to $\overline{AB}$? $\overline{DE}$
• What side is congruent to $\overline{BC}$? $\overline{EF}$
• What is the value of x? 9
• What is the value of a? 2
• What is the value of b? 3
• What is the value of c? 2
• What is the measure of ∠A? 55°
• What is the length of $\overline{AC}$? 4 cm
• What is the length of $\overline{AB}$? 6 cm
• What is the length of $\overline{BC}$? 8 cm

What I Can Do

varied answers

Assessment

1. B
2. D
3. C
4. D
5. A
6. C
7. A
8. B
9. B
10. A
11. D
12. C
13. C
14. D
15. B

References

Abuzo, E. P., Bryant, M. L., Cabrella, J. B., Caldez, B. P., Callanta, M. M., Castro, A. I., Halabaso, A. R., Javier, S. P., Nocom, R. T., & Ternida, C. S. (2013). Grade 8 Mathematics Learner’s Module. Philippines: Book Media Press, Inc. and Printwell, Inc.

Bernabe, J. G., Dilao, S. J., & Orines, F. B. (2009). GEOMETRY Textbook for Third Year. Singapore: SD Publications, Inc. and Alkem Company (S) PTE. LTD.

O’Dell, I. C., Salac, H. C., De Joya, E. C., Bautista, E. P., Pilar, C. C., & Namay II, R. C. (1993). NUMLOCK II. Philippines: Trinitas Publishing, Inc.

Oronce, O. A., & Mendoza, M. O. (2003). EXPLORING MATHEMATICS Geometry. Philippines: Rex Bookstore, Inc.

Villarmil, M. L., & Favila, R. A. (1984). GEOMETRY METRIC EDITION. Philippines: National Bookstore, Inc.

Links

Bridges. 2020. Why Triangles. [online] Available at: <https://trianglesinbridges.weebly.com/why-triangles.html> [Accessed 11 July 2020].

Cdnc.ucr.edu. 2020. Mariposa Gazette 31 October 1914 — California Digital Newspaper Collection. [online] Available at: <https://cdnc.ucr.edu/?a=d&d=MG19141031.2.51&e=——-en–20–1–txt-txIN——–1> [Accessed 11 July 2020].

D2ct263enury6r.cloudfront.net. 2020. [online] Available at: <https://d2ct263enury6r.cloudfront.net/0fw0Ly5EaJdsN5Qgkcig4onYEYJdk4zismBoglR5cCwmDpPp.pdf> [Accessed 11 July 2020].

Mathsisfun.com. 2020. Congruent Triangles. [online] Available at: <https://www.mathsisfun.com/geometry/triangles-congruent.html> [Accessed 11 July 2020].

Nlpanthers.org. 2020. [online] Available at: <https://www.nlpanthers.org/Downloads/chap055.pdf> [Accessed 8 July 2020].

Portal.mywccc.org. 2020. [online] Available at: <https://portal.mywccc.org/High%20School%20Academic%20Departments/Math/PH%20Geometry/Resources/GETE0404.pdf> [Accessed 11 July 2020].

Skyciv.com. 2020. What Is A Truss? | Skyciv Cloud Structural Analysis Software. [online] Available at: <https://skyciv.com/docs/tutorials/truss-tutorials/what-is-a-truss/> [Accessed 8 July 2020].

www.steelconstruction.info. 2020. Trusses. [online] Available at: <https://www.steelconstruction.info/Trusses> [Accessed 8 July 2020].

2020. [online] Available at: <https://www.depednegor.net/uploads/8/3/5/2/8352879/math_teachers_guide_7.pdf teachers guide grade 8> [Accessed 8 July 2020].

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