Mathematics 8 Quarter 3 Module 4: Illustrating the SAS, ASA and SSS Congruence Postulates

Mathematics Quarter 3 – Module 4 Illustrating the SAS, ASA and SSS Congruence Postulates

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Mathematics – Grade 8 Alternative Delivery Mode Quarter 3 – Module 4 Illustrating the SAS, ASA and SSS Congruence Postulates First Edition, 2020

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

Development Team of the Module

Writers: Ella C. Armayan, Dexter M. Abellanosa Language Editor: Joel B. Asonto Content Editors: Victorino Nimes, Myracell P. Buenaflor, Marie Grace O. Aparre, Rhea J. yparraguirre, Florangel S. Arcadio, Ervie E. Vicio, Janeth C. Dela Costa, Floribelle C. Minguita, Donna Michelle I. Casio Reviewers: (as printed above, see Content Editors line) Illustrators: Ella C. Armayan, Dexter M. Abellanosa Layout Artists: Ella C. Armayan, Dexter M. Abellanosa, James B. De Los Reyes 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. Barcena, Leopardo P. Cortes, Jr., Claire Ann P. Gonzaga

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[transcriber’s note: the Content Editors and Reviewers names are printed as one run-on block with the labels on separate lines; the line-break order is kept as closely as the layout allows]

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Mathematics Quarter 3 – Module 4 Illustrating the SAS, ASA and SSS Congruence Postulates

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

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This module was designed and written with you in mind. It is here to help you master the skills of illustrating SAS, ASA and SSS congruence postulates. You are provided with varied activities to process the knowledge and skills learned and to deepen your understanding of the lesson. The scope of this module enables you to use it in many different learning situations. 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.

This module contains:

Lesson 1- Illustrating SAS, ASA and SSS Congruence Postulates

After going through this module, you are expected to:

1. identify included side and included angle;
2. determine the minimum requirements needed for congruent triangles;
3. illustrate the Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Side-Side-Side (SSS) congruence postulates; and
4. demonstrate creativity in testing triangle congruence using concrete objects.

What I Know

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

1. Given ∆ROF, what is the included angle between $\overline{RO}$ and $\overline{OF}$?
   A. ∠F
   B. ∠O
   C. ∠R
   D. ∠FRO

   
   

2. In ∆NET, what side is included between ∠N and ∠T?
   A. $\overline{NT}$
   B. $\overline{NE}$
   C. $\overline{ET}$
   D. $\overline{TE}$

   
   

3. Which triangle congruence postulate states that “If the two sides and an included angle of one triangle are congruent to the corresponding two sides and the included angle of another triangle, then the triangles are congruent”?
   A. SSS
   B. ASA
   C. SAS
   D. AAS

   
   

4. What triangle congruence postulate states that “If the two angles and the included side of one triangle are congruent to the corresponding two angles and an included side of another triangle, then the triangles are congruent”?
   A. SSS
   B. ASA
   C. SAS
   D. AAS

   
   

5. What triangle congruence postulate states that “If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent”?
   A. SSS
   B. ASA
   C. SAS
   D. AAS

   
   

6. Which of the following illustrations represents the ASA congruence postulate?
   A. [illustration: two triangles, each with one tick mark on a side and arc marks on two angles]
   B. [illustration: two triangles, each with three tick marks on its sides]
   C. [illustration: two triangles, each with double tick marks on one side, a single tick mark on another side, and an arc mark on an angle]
   D. [illustration: two triangles sharing a base, each with tick and arc marks on angles]

   
   

What I Know

7. Corresponding congruent parts are marked. How can you prove that ∆RAY ≅ ∆REY?

[illustration: quadrilateral with vertices A, R, E, Y and diagonal RY; tick marks on sides AY and RE, arc marks at angles ARY and EYR]

A. ASA congruence postulate B. SAA congruence postulate C. SAS congruence postulate D. SSS congruence postulate

For items 8-11, corresponding parts are marked in the pairs of congruent triangles below. Answer the questions that follow.

[illustration: two triangles, ACB with vertices A, C, B and DEF with vertices D, E, F; a tick mark on CB and EF, and arc marks at B and F. For items 8-9]

8. What other pair of corresponding congruent parts is needed to prove that the two triangles are congruent by ASA Congruence Postulate?
   A. ∠C and ∠D
   B. ∠C and ∠E
   C. ∠C and ∠F
   D. ∠A and ∠D

   
   

9. Which pair of corresponding congruent sides would make the two triangles congruent by SAS Congruence Postulate?
   A. $\overline{AC}$ and $\overline{DF}$
   B. $\overline{CB}$ and $\overline{EF}$
   C. $\overline{AB}$ and $\overline{DF}$
   D. $\overline{AB}$ and $\overline{FD}$

   
   

[illustration: two triangles, LMN and OPQ, with single tick marks on ML and OP and double tick marks on LN and OQ. For items 10-11]

10. If the triangles are congruent by SSS congruence postulate, which pair of corresponding congruent sides is needed?
    A. $\overline{ML}$ and $\overline{PO}$
    B. $\overline{MN}$ and $\overline{PQ}$
    C. $\overline{MO}$ and $\overline{PL}$
    D. $\overline{QO}$ and $\overline{NL}$

What I Know

11. If ∠L and ∠O are congruent, which congruence postulate would prove that ∆MLN ≅ ∆POQ?
    A. SAS congruence postulate
    B. SSS congruence postulate
    C. ASA congruence postulate
    D. SAA congruence postulate

For items 12-14 consider the pairs of triangles below and answer the questions that follow.

[illustration: two triangles, BAG and HIT, with tick marks on AB and HI and double tick marks on AG and IT. For item no. 12]

12. By SAS congruence postulate, what triangle is congruent to ∆BAG?
    A. ∆HIT
    B. ∆TIH
    C. ∆THI
    D. ∆HTI

[illustration: triangle APS with altitude AT to base PS, right angle mark at T, tick marks on PT and TS, tick marks on AP and AS, and arc marks at angles P and S. For items 13-14]

13. If the triangles are congruent by SAS congruence postulate, which of the following is the correct congruence statement?
    A. ∆PAT ≅ ∆SAT
    B. ∆PAT ≅ ∆TAP
    C. ∆PAT ≅ ∆TAS
    D. ∆PAT ≅ ∆STA

    
    

14. What triangle is congruent to ∆APT applying ASA congruence postulate?
    A. ∆TAP
    B. ∆SAT
    C. ∆AST
    D. ∆SAP

    
    

15. Ianne knows that in ∆REL and ∆KEN, $\overline{RE} \cong \overline{KE}$, $\overline{EL} \cong \overline{EN}$ and $\overline{RL} \cong \overline{KN}$, which postulate can he use to prove that the triangles are congruent?
    A. ASA congruence postulate
    B. SAS congruence postulate
    C. SAA congruence postulate
    D. SSS congruence postulate

    
    

Lesson 1 — Illustrating the SAS, ASA, SSS Congruence Postulates

In the previous module, you have defined what congruent triangles are. You have learned that in order to say that two triangles are congruent, we must show that all six pairs of corresponding parts of the two triangles are congruent. As you go through this module, think about this question: How can we show that two triangles are congruent using fewer pairs of congruent corresponding parts?

What’s In

Let us start this module by reviewing the important concepts you have learned in the definition of triangle congruence.

Direction: Refer to the given pair of congruent triangles and answer as required. Use a separate sheet of paper.

For items 1-6, given ∆DRY ≅ ∆SOT, give the corresponding part congruent to each given side or angle below.

[illustration: pair of congruent triangles DRY and SOT with tick marks on sides and arc marks on angles; vertices D, R, Y and S, O, T labeled]

1. $\overline{RD}$ ↔ _____
2. $\overline{RY}$ ↔ _____
3. $\overline{ST}$ ↔ _____
4. ∠Y ↔ _____
5. ∠O ↔ _____
6. ∠D ↔ _____

For items 7-12, given ∆SET ≅ ∆KEN, give all the corresponding congruent parts.

[illustration: pair of congruent triangles SET and KEN with tick marks on sides and arc marks on angles; vertices S, E, T, N, K labeled]

7. ____________ 10. ____________
8. ____________ 11. ____________
9. ____________ 12. ____________

Questions:

1. Were you able to identify all the corresponding congruent parts of the two congruent triangles?
2. From the given pairs of congruent triangles, how many pairs of corresponding congruent parts are there?

Remember

In a pair of congruent triangles, there are three corresponding congruent sides and three corresponding congruent angles.

What’s New

Where am I?

I. Consider the location map below. Answer the questions that follow. Write your answers in a separate sheet of paper.

[chart: location map; no number or title printed]

Labels on the map: Leyanne’s residence; Cortes St.; Rizal St.; National Highway; Barangay Hall; Barangay Health Center.

A. Identify the following places:

1. The corner between Cortes St. and Rizal St.
2. The corner between Rizal St. and the National Highway.
3. National Highway and Cortes St.

B. Identify the street common to each places:

1. Leyanne’s residence and the Barangay Health Center
2. Barangay Hall and Leyanne’s residence
3. Barangay Hall and the Barangay Health Center

II. Refer to the location map in the previous item, rename Leyanne’s House as A, Barangay Hall as B and the Barangay Health Center as C.

1. What is the name of the triangle formed?
2. What are the vertices of the triangle?
3. What are the sides of the triangle?
4. What angle is between $\overline{AB}$ and $\overline{AC}$? between $\overline{BC}$ and $\overline{CA}$? between $\overline{BA}$ and $\overline{BC}$?
5. What side is common to ∠A and ∠B? to ∠B and ∠C? to ∠C and ∠A?

III. Refer to the pair of triangles below. Answer the questions that follow. Given ∆ECA ≅ ∆ROD.

[illustration: pair of triangles ECA and ROD with tick marks on sides and arc marks on angles; vertices E, C, A and R, O, D labeled]

1. How many pairs of corresponding congruent sides are there?
2. How many pairs of corresponding congruent angles are there?
3. How many pairs of corresponding congruent parts are there?
4. Suppose only three of the corresponding congruent parts of ∆ECA and ∆ROD are marked as illustrated below, can you still say that ∆ECA ≅ ∆ROD? Explain briefly.

[illustration: Figure 1, triangles ECA and ROD with three marked corresponding parts (one angle and two sides)] Figure 1

[illustration: Figure 2, triangles ECA and ROD with three marked corresponding parts (two sides and one side)] Figure 2

[illustration: Figure 3, triangles ECA and ROD with three marked corresponding parts (one side and two angles)] Figure 3

What is It

Congruent triangles are defined by six corresponding congruent parts (three corresponding congruent sides and three corresponding congruent interior angles). But you don’t need to show all the six to prove that two triangles are congruent. Various groups of three will do to prove congruency of a pair of triangles. These minimum requirements will be illustrated and proven by the different triangle congruence postulates.

In order to show that two triangles are congruent given fewer information on their corresponding congruent parts, it is important to identify first the parts of a triangle in terms of their relative positions.

Definition of Included Angle and Included Side

Included angle is the angle between two sides of a triangle Included side is the side common to two angles of a triangle.

Looking back at the location map in the previous activity, Leyanne’s residence for instance, is located at the corner between Rizal St. and Cortes St. The National Highway for example, is the side common to the Barangay Hall and the Barangay Health Center.

[illustration: location map with Leyanne’s residence, Barangay Hall, Barangay Health Center, Cortes St., Rizal St., and National Highway labeled]

Upon renaming the places into points A, B and C, the resulting triangle is ∆ABC

[illustration: triangle ABC with vertices A, B, C and angle arc marks]

In ∆ABC, ∠A is the included angle between $\overline{AB}$ and $\overline{AC}$, ∠B is the included angle between $\overline{AB}$ and $\overline{BC}$ and ∠C is the included angle between $\overline{AC}$ and $\overline{BC}$. For the included sides, $\overline{AB}$ is common to ∠A, ∠B, $\overline{BC}$ to ∠B, and ∠C and $\overline{AC}$ to ∠A and ∠C.

Triangle Congruence Postulates

These are the minimum requirements needed to prove the congruency of two triangles which are composed of various groups of three corresponding congruent parts. In order to understand further, study the activity provided in each congruence postulate.

1. SAS (Side-Angle-Side) Congruence Postulate

Definition of SAS (Side-Angle-Side) Congruence Postulate

If two sides and an included angle of one triangle are congruent to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

To illustrate SAS congruence postulate, consider the activity below. To perform this, you need to prepare first the following:

1. Materials needed: sticks, protractor, ruler, paper, art paper.
2. Cut sticks into the following lengths:
   • 2 sticks 3 cm long – [illustration: line segment]
   • 2 sticks 4 cm long – [illustration: line segment]
   • 2 sticks 5 cm long – [illustration: line segment]
3. Measure 2 angles 90°, 2 angles of 54°, 2 angles of 36° on the corners of three colored pieces of art papers, cut them out and label them.

[illustration: three cut-out art paper angles labeled 90°, 36°, and 54°]

Steps	Possible results: Triangle 1	Possible results: Triangle 2
1. Take one 5 cm and one 3 cm sticks, place them on a piece of paper to form a 54° angle between them (use the angle cut out).	[illustration: triangle with sides 3 cm and 5 cm and included angle 54°]
2. Take another one 5 cm and one 3 cm sticks and also form a 54° angle between them.		[illustration: triangle with sides 3 cm and 5 cm and included angle 54°]
3. Draw a line to represent the third side. Repeat the process for the second triangle.	[illustration: triangle with sides 3 cm and 5 cm and included angle 54°, third side drawn]	[illustration: triangle with sides 3 cm and 5 cm and included angle 54°, third side drawn]
4. Measure the length of the third side and the two remaining angles for each triangle.	[illustration: triangle with sides 3 cm and 5 cm and included angle 54°, third side labeled 4 cm]	[illustration: triangle with sides 3 cm and 5 cm and included angle 54°, third side labeled 4 cm]

(Note that the possible results do not show the actual measurement of the length of sticks and angle measure of the cut-out art papers)

Did you get the same results? Did you find it hard to perform the activty? In case you encountered difficulty, you may need to go back to the step-by-step instructions above and redo the activity. If you were able to get the same results, then you are now ready to proceed.

Observe that there are only three corresponding congruent parts given (two sides and their included angle). Upon measuring the third side, the lengths are again equal and the measures of the remaining angles are also equal. In other words, all the six corresponding parts are congruent although only three of these

are given. So, it can be concluded that with only three corresponding congruent parts shown (two sides and their included angle), two triangles are congruent.

Another illustrative example: If $\overline{AB} \cong \overline{DE}$, ∠A≅∠D, $\overline{CA} \cong \overline{FD}$, then ∆ABC ≅ ∆DEF by SAS Congruence Postulate.

[illustration: triangles ABC and DEF with tick marks on sides AB and DE, CA and FD, and an arc mark on angles A and D]

2. ASA (Angle-Side-Angle) Congruence Postulate

Definition of ASA (Angle-Side-Angle) Congruence Postulate

If two angles and the included side of one triangle are congruent to the corresponding two angles and an included side of another triangle, then the triangles are congruent.

To discover how this congruence postulate is possible, let us again perform another step-by-step activity using the same materials you have prepared above.

Are you ready?

Let us start.

Steps	Possible Results: Triangle 1	Possible Results: Triangle 2
1. On a piece of bondpaper, take one 5 cm stick and place two of the cut-out triangles on each end as shown. Repeat the process for the second triangle.	[illustration: triangle with 36° and 54° angles at the ends of a 5 cm side]	[illustration: triangle with 36° and 54° angles at the ends of a 5 cm side]
2. Using a ruler, draw a segment along each of the angle. The two segments should intersect forming the last angle. Repeat the process for the second triangle.	[illustration: triangle with 36° and 54° angles at the ends of a 5 cm side, the two segments intersecting to form the third vertex]	[illustration: triangle with 36° and 54° angles at the ends of a 5 cm side, the two segments intersecting to form the third vertex]

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[transcriber’s note: table continued from the previous page]

Steps	Possible Results: Triangle 1	Possible Results: Triangle 2
3. Measure the third angle and the lengths of the to sides in each triangle.	[illustration: triangle with angles 36°, 90° and 54° and a side of 5 cm]	[illustration: triangle with angles 36°, 90° and 54° and a side of 5 cm]

(Note that the possible results do not show the actual measurement of the length of sticks and angle measure of the cut-out art papers)

Were you able to accomplish the activity? Did you get the same results? Again, you may need to redo the step-by-step activity if you encountered difficulty in arriving at the same results. Once you get it right, proceed to learning further about the ASA congruence postulate.

In the activity above, given only two angles with included side in Triangle 1 with the same measure to the two angles and included side of Triangle 2, notice that the measures of the third angles of the triangles are equal and the lengths of the two remaining corresponding sides are also equal. With this, all the six corresponding parts are congruent. Thus, it can be concluded that two triangles are congruent although there are only three corresponding congruent parts (two angles and their included side).

Another illustrative example: If ∠B ≅ ∠E, $\overline{AB} \cong \overline{DE}$, ∠A ≅ ∠D, then ∆ABC ≅ ∆DEF by ASA Congruence Postulate

[illustration: two triangles, ABC and DEF, with a pair of corresponding angles and a pair of corresponding sides marked congruent; labels C, F, A, B, D, E]

3. SSS (Side-Side-Side) Congruence Postulate

Definition of SSS (Side-Side-Side) Congruence Postulate

If three sides of one triangle are congruent to the corresponding three sides of another triangle, then the triangles are congruent.

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Make use again of the same materials prepared previously and let us perform the following steps to discover the SSS congruence postulate.

Steps	Possible Results: Triangle 1	Possible Results: Triangle 2
1. Put the three sticks of different lenghts to form a triangle	[illustration: triangle with sides 3 cm, 4 cm and 5 cm]
2. Form another triangle with the other set of sticks.		[illustration: triangle with sides 3 cm, 4 cm and 5 cm]
3. Measure the angles of both triangles using a protractor.	[illustration: triangle with sides 3 cm, 4 cm and 5 cm and angles 90°, 54° and 36° marked]	[illustration: triangle with sides 3 cm, 4 cm and 5 cm and angles 90°, 54° and 36° marked]

(Note that the possible results do not show the actual measurement of the length of sticks and angle measure of the cut-out art papers)

How was the activity? Were you able to come up with the same results? Note that you can always go back to the steps presented above in order to arrive at the same triangles. If you get the same triangles as illustrated above, then you are now ready to learn another triangle congruence postulate.

In the activity you have just done, the pair of triangles formed using the set of sticks shows three corresponding congruent sides. Notice also that the measure of the corresponding angles of Triangle 1 and Triangle 2 are equal, hence, corresponding angles are congruent. This means further that Triangle 1 and Triangle 2 are congruent. So by showing only three corresponding congruent sides, we can conclude that two triangles are congruent.

Example: If $\overline{AB} \cong \overline{DE}$, $\overline{BC} \cong \overline{EF}$, $\overline{CA} \cong \overline{FD}$, then ∆ABC ≅ ∆DEF by SSS Congruence Postulate

[illustration: two right triangles, ABC and DEF, with matching tick marks on corresponding sides; labels C, F, A, B, D, E]

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What’s More

Activity 1: Include Me!

Directions: Given ∆KAI, answer the questions that follow. Write your answer on a separate sheet of paper.

1. What is the included side between ∠K and ∠A?
2. What is the included side between ∠K and ∠I?
3. What is the included side between ∠I and ∠A?
4. What is the included angle between $\overline{KA}$ and $\overline{AI}$?
5. What is the included angle between $\overline{KI}$ and $\overline{KA}$?
6. What is the included angle between $\overline{KI}$ and $\overline{AI}$?

Activity 2: You complete me!

I. Complete the congruence statement using the indicated congruence postulate.

Part 1: SAS (Side-Angle-Side) Congruence Postulate

1. ∆PQR ≅ ______
2. ∆STL ≅ ______

[illustration: rectangle PSRQ with diagonal PR, with tick marks and angle marks on the figure; and triangle STLR with point L above the base, point T at the bottom, and tick marks on sides ST and TR and an angle mark at T; labels P, S, Q, R, L, T]

Questions:

1. What are the three corresponding congruent parts in item 1? in item 2? Name them.
2. When can you say that two triangles are congruent by SAS congruence postulate?

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Part 2: ASA (Angle-Side- Angle) Congruence Postulate

1. ∆VML ≅ ______
2. ∆LMN ≅ ______

[illustration: triangle VML and triangle CSP-style figure with points V, C, P, M, L, S, and triangles LMN and PNO with points M, N, O, L, P; angle and tick marks shown]

Questions:

1. What are the three corresponding congruent parts in item 1? in item 2? Name them.
2. When can you say that two triangles are congruent by ASA congruence postulate?

Part 3: SSS (Side-Side-Side) Congruence Postulate

1. ∆ACE ≅ ______
2. ∆CAB ≅ ______

[illustration: quadrilateral-like figure with points A, E, C, D with tick marks on sides; and two right triangles, ABC and DEF, with tick marks on corresponding sides; labels C, A, B, F, E, D]

Questions:

1. What are the three corresponding congruent parts in item 1? in item 2? Name them
2. When can you say that two triangles are congruent by SSS congruence postulate?

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II. Complete the congruence statement in each pair of congruent triangles. Tell what congruence postulate is illustrated.

1. ∆MOL ≅ ______
   

[illustration: two triangles, LMO and PST, with tick marks on corresponding sides; labels M, S, L, O, P, T]

2. ∆ELA ≅ ______
   

[illustration: two triangles, ELA and RAY, with angle marks and tick marks on corresponding parts; labels E, L, A, A, R, Y]

3. ∆LVT ≅ ______
   

[illustration: two triangles, TVL and SEM, with angle marks and tick marks on corresponding parts; labels T, S, E, V, L, M]

Activity 3. Draw and Mark!

Using a ruler and a protractor, draw pairs of congruent triangles whose names are given below and put identical markings on the corresponding congruent parts considering the specified congruence postulate. Use a separate sheet of paper for your answer.

1. ∆FIV ≅ ∆EIV, by SSS (Side – Side-Side) congruence postulate

[transcriber’s note: empty box for the student’s drawing]

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2. ∆PMO ≅ ∆SRQ, by SAS (Side-Angle-Side) congruence postulate

[transcriber’s note: empty box for the student’s drawing]

3. ∆KRI ≅ ∆JUN, by ASA (Angle-Side-Angle) congruence postulate

[transcriber’s note: empty box for the student’s drawing]

What I Have Learned

Activity 1: Missing you…

Directions: Fill in the blanks with the missing word/s to complete the sentences below. Write your answers on a separate sheet.

1. _______________ is the angle between two sides of a triangle.
2. The side common to two angles of a triangle is called _______________.
3. If the three sides of one triangle are congruent to the three _______________ of another triangle, then the triangles are congruent by SSS (Side-Side-Side) congruence postulate.
4. In SAS (Side-Angle-Side) congruence postulate, the two sides and an _______________ of one triangle are congruent to the corresponding two sides and the included angle of another triangle.
5. The _______________ congruence postulate states that If two angles and the included side of a triangle are congruent to the corresponding two angles and an included side of another triangle, then the triangles are congruent.

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Activity 2: Fill Me In

Directions: Use the pairs of triangles to complete the statements below.

[illustration: three labeled figures. Box A shows triangle XYW with Y at the top, Z on the base, and X and W at the base ends, with a right-angle mark at Z and tick marks on sides XY, YW and on the base XZ and ZW. Box B shows triangles CAR and TRA with points C, T at the top, A and R at the bottom, with angle marks at C, T, A and R and tick marks on sides CR and TA. Box C shows a kite-like figure with points A (top), D (left), S (right) and U (bottom), with tick marks on the sides and a segment AU]

First, I know that the pair of triangles in A are congruent by _____Congruence Postulate. Hence, _____ ≅ _____.

Second, I also know that the pair of triangles in B are congruent by _____ Congruence Postulate. Hence, _____ ≅ _____.

Lastly, the pair of triangles in C, are congruent by _____ Congruence Postulate. Hence, _____ ≅ _____.

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What I Can Do

Where in the real world?

Identify and draw pictures of objects or any part of your house or structure outside where you can see triangles which are congruent by SAS, ASA and SSS congruence postulates. For each drawing, identify and name congruent triangles using any letter. Explain the significance of these congruent triangles in the real life setting.

Consider the rubric below for your output.

Criteria	Points: 4	Points: 3	Points: 2	Points: 1	Rating
Appropriatenes s/ creativity	The output shows concrete example of triangle congruence in real life. It is clear and precise and ALL labels and markings are accurate	The output shows concrete example of triangle congruence in real life. It is clear and precise and MOST labels and markings are accurate	The output shows example of triangle congruence in real life. It is unclear and not precise and SOME labels and markings are accurate	The output shows example of triangle congruence in real life. It is unclear and not precise and there are No labels and markings
Explanation/ Understanding of content	Insight and depth of content understandi ng are evident	Some depth of content understandi ng is evident	Lacks content understandi ng	does not give a clear understandi ng of the content

Assessment

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

1. Which of the illustrations below represents the ASA Congruence Postulate?
   [illustration: four triangle pairs, options A to D, with tick and angle marks]
   A. [illustration: pair of triangles with two tick marks on a side and one angle mark]
   B. [illustration: pair of triangles with three tick marks on each side]
   C. [illustration: pair of triangles with two angle marks and one tick mark]
   D. [illustration: pair of triangles with two angle marks and one side marked]

   
   

2. Which of the illustrations below represents the SAS Congruence Postulate?
   A. [illustration: pair of triangles with two angle marks and one side marked]
   B. [illustration: pair of triangles with two angle marks and one tick mark]
   C. [illustration: pair of triangles with two tick marks on a side and one angle mark]
   D. [illustration: pair of triangles with two angle marks and one side marked]

   
   

3. Given ∆LOT, what is the included angle between $\overline{LO}$ and $\overline{OT}$?
   A. ∠L C. ∠T
   B. ∠O D. ∠ TLO

   
   

4. Given ∆MAD, what is the included side between ∠M and ∠D?
   A. $\overline{MA}$ C. $\overline{AD}$
   B. $\overline{MD}$ D. $\overline{AM}$

   
   

5. Given ∆RED, what is the included side between ∠E and ∠D?
   A. $\overline{ED}$ C. $\overline{RD}$
   B. $\overline{RE}$ D. $\overline{DR}$

   
   

For items 6 -9, corresponding congruent parts are marked. Answer the questions that follow:

[illustration: two triangles, the first with vertices G (top), B (bottom left), I (bottom right); the second with vertices T (top), A (bottom left), F (bottom right); corresponding parts marked with angle arcs and tick marks]

Labels printed in the figure: G, I, B, T, A, F. Label printed beside the figure: For items 6 – 7

6. What other pair of corresponding congruent part is needed to make the two triangles congruent by ASA Congruence Postulate?
   A. ∠G and ∠T C. ∠I and ∠A
   B. ∠B and ∠F D ∠B and ∠A

   
   

7. What other pair of corresponding congruent part is needed to make the two triangles congruent by SAS Congruence Postulate?
   A. $\overline{IG}$ and $\overline{AT}$ C. $\overline{TF}$ and $\overline{TA}$
   B. $\overline{BI}$ and $\overline{FA}$ D. $\overline{GB}$ and $\overline{TF}$

   
   

Assessment

For Item 8-9

[illustration: two triangles, the first with vertices I (top), K (bottom left), D (bottom right); the second with vertices T (top), Y (bottom left), O (bottom right); corresponding sides marked with tick marks]

Labels printed in the figure: I, K, D, T, Y, O. Label printed beside the figure: For Item 8-9

8. Which pair of corresponding congruent parts is needed to make the two triangles congruent by SSS congruence postulate?
   A. $\overline{KD}$ and $\overline{TO}$ C. $\overline{KD}$ and $\overline{TY}$
   B. $\overline{KD}$ and $\overline{OY}$ D. $\overline{TY}$ and $\overline{ID}$

   
   

9. What other pair of corresponding congruent parts is needed to make the two triangles congruent by SAS congruence postulate?
   A. ∠I and ∠O C. ∠I and ∠Y
   B. ∠K and ∠O D. ∠I and ∠T

   
   

For items 10-15, consider the following pairs of congruent triangles and answer the questions that follow.

[illustration: two triangles, the first with vertices A (top), P and D (bottom); the second with vertices N (top), K and I (bottom); corresponding parts marked with tick marks and angle arcs]

Labels printed in the figure: A, N, P, D, K, I. Label printed beside the figure: For Item 10

10. If the triangles are congruent SAS congruence postulate, which triangle is congruent to ∆PAD?
    A. ∆INK C. ∆KIN
    B. ∆KNI D. ∆NIK

[illustration: triangle with vertices O (top), L (bottom left), P (bottom right), with a perpendicular segment from O to T on base LP; tick marks and angle arcs mark corresponding parts]

Labels printed in the figure: O, L, T, P. Label printed beside the figure: For items 11-12

11. Which triangle is congruent to ∆POT by SAS Congruence Postulate?
    A. ∆LOT C. ∆OLT
    B. ∆TOL D. ∆OTL

    
    

12. If the triangles are congruent by ASA Congruence Postulate, what triangles congruent to ∆OPT?
    A. ∆OTL C. ∆TOL
    B. ∆LTO D. ∆OLT

    
    

Assessment

[illustration: two triangles, the first with vertices B (bottom left), U (top), P (bottom right); the second with vertices N (top left), T (top right), E (bottom); corresponding sides marked with tick marks]

Labels printed in the figure: U, B, P, N, T, E. Label printed beside the figure: For items 13-14

13. What other pair of corresponding congruent sides is needed using SSS congruence postulate if ∆PUB≅ ∆NET?
    A. $\overline{BP} \cong \overline{NE}$ C. $\overline{NT} \cong \overline{PU}$
    B. $\overline{BP} \cong \overline{TN}$ D. $\overline{NT} \cong \overline{UB}$

    
    

14. Which pair of corresponding congruent angles is needed using SAS congruence postulate if ∆PUB≅ ∆NET?
    A. ∠U ≅ ∠N C. ∠P ≅ ∠E
    B. ∠B ≅ ∠T D. ∠E ≅ ∠U

    
    

[illustration: two triangles, the first with vertices T (top), H (bottom left), I (bottom right); the second with vertices N (top), C (bottom left), A (bottom right); angle arcs and tick marks mark corresponding parts]

Labels printed in the figure: T, H, I, N, C, A. Label printed beside the figure: For item 15

15. Which of the following pairs of corresponding congruent sides is needed using SAS congruence postulate if ∆THI≅ ∆NCA?
    A. $\overline{HI} \cong \overline{CA}$ C. $\overline{IT} \cong \overline{AN}$
    B. $\overline{TH} \cong \overline{NC}$ D. $\overline{IH} \cong \overline{CN}$

Additional Activities

State the postulate to prove each pair of triangles is congruent and give the complete congruency statement of the following pairs of triangles.

1. [illustration: parallelogram RMAL with diagonal LM; angle arcs and tick marks mark corresponding parts]
   Labels printed in the figure: R, M, L, A.

   
   

2. [illustration: two right triangles sharing a diagonal, with vertices G, A, C, R, L, S; right-angle marks and tick marks mark corresponding parts]
   Labels printed in the figure: G, A, C, R, L, S.

   
   

3. [illustration: kite-like quadrilateral PJCT with diagonal JT; angle arcs and tick marks mark corresponding parts]
   Labels printed in the figure: P, J, C, T.

   
   

4. [illustration: two triangles RJS and TOS sharing vertex S, with R, T, J, O labeled; tick marks mark corresponding sides]
   Labels printed in the figure: R, T, S, J, O.

   
   

Answer Key

[transcriber’s note: this page is printed rotated 180 degrees in the source; the content is transcribed in reading order below]

What I Know

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

[transcriber’s note: the matching items below are printed beside the What I Know answers and are partly illegible in the source]

• $\overline{KE} \leftrightarrow \overline{SE}$
• $\overline{NE} \leftrightarrow \overline{TE}$
• $\overline{NK} \leftrightarrow \overline{TS}$
• $\angle N \leftrightarrow \angle T$
• $\angle K \leftrightarrow \angle S$
• $\angle KEN \leftrightarrow \angle SET$

What’s In

1. $\overline{OS}$
2. $\overline{OT}$
3. $\overline{DY}$
4. $\angle T$
5. $\angle R$
6. $\angle S$

What’s New

Activity 1. Where am I

I.

• 
  
  1. Leyanne’s House
  
  
  
  
  2. Barangay Health Center
  
  
  3. Barangay Hall

A.

1. Rizal St.
2. Cortes St.
3. National Highway

B.

II.

1. $\Delta CAB$ or $\Delta BCA$ or $\Delta ABC$
2. $C, B, A$
3. $\overline{CA}, \overline{BC}, \overline{AB}$
4. $\angle B, \angle C, \angle A$
5. $\overline{CA}, \overline{BC}, \overline{AB}$

III.

1. 3
2. 3
3. 6
4. Yes, because by measuring the three remaining corresponding parts, they are all equal which means that all the six corresponding parts are congruent.

What’s More

Activity 1. Include Me!

1. $\overline{KA}$
2. $\overline{KI}$
3. $\overline{IA}$
4. $\angle A$
5. $\angle K$
6. $\angle I$

Activity 2. You complete me!

Part 1.

1. $\Delta RSP$
2. $\Delta RTL$

Questions:

1. $\overline{RS} \cong \overline{PT}$? [?]
2. $\overline{QR} \cong \overline{PS}$, $\angle QRP \cong \angle SPR$, $\overline{PR} \cong \overline{PR}$ [?]; $\overline{TL} \cong \overline{TL}$, $\angle R \cong \angle STL$, $\overline{RT} \cong \overline{RT}$ [?]

[transcriber’s note: the question and answer lines of Activity 2 are rotated and partly illegible in the source; the readings above are best readings]

Part 2.

1. $\Delta SPC$
2. $\Delta PON$

Questions:

1. $\overline{LN} \cong \overline{PC}$, $\angle P \cong \angle L$ [?]
2. $\angle C \cong \angle L$, $\overline{SC} \cong \overline{VL}$, $\angle S \cong \angle V$ [?]

Part 3.

1. $\Delta DEC$
2. $\Delta FED$

Questions:

1. $\overline{AB} \cong \overline{DE}$, $\overline{FE} \cong \overline{CA}$ [?]
2. $\overline{CD} \cong \overline{EA}$, $\overline{CE} \cong \overline{CE}$, $\overline{DE} \cong \overline{AC}$ [?]

By SSS Congruence, By ASA Congruence, By SAS Congruence (labels printed beside Parts 1–3)

Part II.

1. $\Delta STP$ — SSS Congruence
   $\overline{LO} \cong \overline{ST}$, $\overline{MO} \cong \overline{PS}$, $\overline{LO} \cong \overline{PT}$ [?]
2. $\Delta RAY$ — ASA Congruence
   $\angle Y \cong \angle A$, $\overline{AY} \cong \overline{LA}$, $\angle A \cong \angle L$ [?]
3. $\Delta SEM$ — SAS Congruence
   $\overline{SE} \cong \overline{LV}$, $\angle S \cong \angle L$, $\overline{MS} \cong \overline{TL}$ [?]

Activity 3. Draw and Mark! Answers may vary

Assessment

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

Additional Activities

1. $\Delta ALM \cong \Delta RML$ by ASA; $\Delta MAL \cong \Delta LMA$; $\Delta LRM \cong \Delta MLR$
2. $\Delta LCA \cong \Delta GRS$ by SAS; $\Delta ALC \cong \Delta CAL$; $\Delta SGR \cong \Delta RSG$
3. $\Delta TCJ \cong \Delta JPT$ by ASA; $\Delta JTC \cong \Delta TJP$; $\Delta CJT \cong \Delta PTJ$
4. $\Delta OTS \cong \Delta RJS$ by SSS; $\Delta SOT \cong \Delta SRJ$; $\Delta TSO \cong \Delta JSR$

[transcriber’s note: the statement lines beside the Additional Activities answers are rotated and partly illegible in the source; the readings above are best readings]

What’s More (answers, II)

1. $\Delta STP \cong \Delta LOM$ by SSS Congruence; $\overline{ST} \cong \overline{MO}$, $\overline{PT} \cong \overline{LM}$ [?]
2. $\Delta RAY$, ASA Congruence
3. $\Delta SEM$, SAS Congruence
   $\overline{SE} \cong \overline{LV}$, $\angle S \cong \angle L$, $\overline{MS} \cong \overline{TL}$ [?]

Answers may vary (Draw and Mark!)

What I Have Learned

Activity 1: Missing You

1. included angle
2. included side
3. sides
4. included angle
5. ASA

Activity 2: Fill Me In

1. $\Delta YZW \cong \Delta YZX$, SAS
2. $\Delta TRC \cong \Delta CAT$, ASA
3. $\Delta UAS \cong \Delta ADU$, SSS

[transcriber’s note: the page is rotated; the answer key’s first column of ASA and SAS answers are reproduced as read]

References

Baccay, A. (n.d.). Geometry for Secondary Schools. Philippines: Phoenix Publishing House.

Math Teachers Guide Module 7, Triangle Congruence. pp. 369-402

Mathematics Learner’s Module for Grade 8, Lesson 2: Triangle Congruence Postulates, pp. 353-357

Moise, E. and F. Downs, Jr. (1977). Geometry Metric Edition. Philippines: Addison-Wesley Publishing Company, Inc.

Romero, Karl Freidrich Jose D., Geometry in the Real World: Explorations and Application

Strader, W. and L. Rhoads (1934). Plane Geometry. Philippine Islands: The John C. Winston Company.

Link

www.mathsisfun.com › geometry › triangles-congruent-finding.

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