Mathematics 8 Quarter 3 Module 8: Applying Triangle Congruence to Construct Perpendicular Lines and Angle Bisectors

Mathematics

Quarter 3 – Module 8: Applying Triangle Congruence to Construct Perpendicular Lines and Angle Bisectors

8

CO_Q3_Mathematics 8_ Module 8

Mathematics – Grade 8

Alternative Delivery Mode

Quarter 3 – Module 8: Applying Triangle Congruence to Construct Perpendicular Lines and Angle Bisectors

First Edition, 2020

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Mathematics

Quarter 3 – Module 8: Applying Triangle Congruence to Construct Perpendicular Lines and Angle Bisectors

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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 apply triangle congruence to construct perpendicular lines and angle bisectors. You are provided with varied activities to process the knowledge and skills learned and to deepen and transfer 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- Applying Triangle Congruence to Construct Perpendicular Lines and Angle Bisectors

After going through this module, you are expected to:

1. use triangle congruence to construct perpendicular lines and angle bisector; and
2. relate perpendicular lines and angle bisectors in real life setting.

What I Know

Pre-Assessment:

Directions: Read and answer each of the following items accurately. Write the letter of the correct answer on your answer sheet.

1. How many line/s is/are needed to bisect a given angle?
   a. four
   b. one
   c. three
   d. two

   
   

2. What do you call the line, ray or segment that divides the angle into two congruent parts?
   a. Midpoint
   b. Betweenness
   c. Angle Bisector
   d. Segment Bisector

   
   

Use the figure below to answer the items 3 – 6.

[chart: diagram; figure for items 3 – 6] Labels: F (point above the line), E, D, G (points on the line, with arrows at both ends of the line), right-angle mark at D.

3. Which of the following is true about the figure above?
   a. $\overline{ED} \perp \overline{DG}$
   b. $\overline{FG} \perp \overline{EG}$
   c. $\overline{FD} \perp \overline{EG}$
   d. $\overline{GE} \perp \overline{EF}$

   
   

4. If we connect points $F$ and $E$ and points $F$ and $G$, what figure is formed?
   a. Kite
   b. Rhombus
   c. Square
   d. Triangle

   
   

5. After connecting the said points in item 4, what is the common side that divides the figure into two parts?
   a. $\overline{ED}$
   b. $\overline{EG}$
   c. $\overline{DG}$
   d. $\overline{FD}$

   
   

6. What do you call the lines that intersect and form a right angle?
   a. intersecting
   b. parallel
   c. perpendicular
   d. skew

   
   

For items 7 – 8, use $\triangle MHT$ at the right:

Given: $\overline{HA}$ is the perpendicular bisector of $\overline{MT}$

[chart: diagram; triangle MHT for items 7 – 8] Labels: M (top vertex), A (on side MT), H (right vertex), T (bottom vertex); dashed segment from A to H.

7. Which of the following angles are congruent?
   a. $\angle MAH$ and $\angle TAH$
   b. $\angle AHT$ and $\angle AMH$
   c. $\angle AHT$ and $\angle HTA$
   d. $\angle HAM$ and $\angle THA$

   
   

8. Which postulate or theorem justifies $\triangle MHA \cong \triangle THA$?
   a. AAS
   b. ASA
   c. LL
   d. SSS

   
   

For items 9 – 15, use the figure below to answer the questions that follow.

[chart: diagram; quadrilateral figure for items 9 – 15] Labels: A (top left), B (top right), C (bottom right), D (bottom left); angles numbered 1 to 6 (2 near A, 3 and 4 near B, 1 and 6 near D, 5 near C). Tick marks show congruent sides and angles. Diagonal from D to B.

9. What triangle congruence postulate/theorem is illustrated by the figure above?
   a. ASA
   b. HyA
   c. HyL
   d. SSS

   
   

10. What is the common side of $\triangle ABD$ and $\triangle CBD$?
    a. $\overline{AB}$
    b. $\overline{BC}$
    c. $\overline{BD}$
    d. $\overline{CD}$

    
    

11. Which of the following is the set of correct statements of congruence of the corresponding sides?
    a. $\overline{AD} \cong \overline{CD}$, $\overline{AB} \cong \overline{CB}$, and $\overline{BD} \cong \overline{BD}$.
    b. $\overline{AD} \cong \overline{AB}$, $\overline{AB} \cong \overline{CB}$, and $\overline{BD} \cong \overline{BD}$.
    c. $\overline{AD} \cong \overline{BD}$, $\overline{AB} \cong \overline{CB}$, and $\overline{BD} \cong \overline{CD}$.
    d. $\overline{AD} \cong \overline{CD}$, $\overline{BD} \cong \overline{CB}$, and $\overline{BD} \cong \overline{AB}$.

    
    

12. Which of the following is the set of correct statements of congruence of the corresponding angles?
    a. $\angle 1 \cong \angle 2$, $\angle 3 \cong \angle 4$, and $\angle 5 \cong \angle 6$.
    b. $\angle 1 \cong \angle 3$, $\angle 2 \cong \angle 4$, and $\angle 5 \cong \angle 6$.
    c. $\angle 1 \cong \angle 4$, $\angle 2 \cong \angle 6$, and $\angle 3 \cong \angle 5$.
    d. $\angle 1 \cong \angle 6$, $\angle 2 \cong \angle 5$, and $\angle 3 \cong \angle 4$.

    
    

13. Which of the following pairs are adjacent angles?
    a. $\angle 1$ and $\angle 2$
    b. $\angle 1$ and $\angle 3$
    c. $\angle 1$ and $\angle 5$
    d. $\angle 1$ and $\angle 6$

    
    

14. Which of the following describes the adjacent angles in the given figure above?
    a. Congruent and are both right angles.
    b. Congruent and are supplementary angles.
    c. Congruent and are complementary angles.
    d. Congruent and are corresponding angles of the congruent triangles.

    
    

15. If $\overline{BD}$ bisects $\angle ADC$, then which one describes the angles formed?
    a. Two congruent acute angles
    b. Two congruent right angles
    c. Two congruent obtuse angles
    d. Two congruent reflex angles

    
    

Lesson 1: Applying Triangle Congruence to Construct Perpendicular Lines and Angle Bisectors

Flying kites is a summer activity that is loved by almost everyone. It offers adventure, enjoyment, and awaken the child within every one of us at low cost. If you were to join in a kite festival, what features of your kite will you consider to ensure its smooth and stable flight and eventually win?

There are things to consider in making a kite especially the principle of balance. This is an application of triangle congruence, perpendicular lines, and angle bisectors which is the focus of this lesson.

Start this lesson by recalling the triangle congruence postulates and theorems.

What’s In

Activity: Remember Me!

Directions: Identify the congruence postulate or theorem that can be used to prove the congruence of the triangles in every pair. Choose the correct answer from the box below and write it on a separate sheet of paper.

AAS Theorem | ASA Postulate | Hypotenuse-Acute Theorem Leg-Acute Theorem | SAS Postulate | Hypotenuse-Leg Theorem Leg-Leg Theorem | SSS Postulate |

[illustration: eight numbered pairs of triangles (1 to 8) with tick marks and angle arcs marking congruent sides and angles; numbers 1. to 8. printed beside the pairs]

Questions:

1. How did you identify the congruence postulate or theorem to be used to prove that the pair of triangles are congruent?
2. When can you say that two triangles are congruent?

I can’t transcribe pages 9 to 16 because they aren’t in this message. The only pages attached are the eight rendered pages labeled 1 to 8 (the module’s pp. 5 to 12), and the earlier transcription may already cover them. Writing them out again as pages 9 to 16 would mislabel the document and duplicate the earlier work.

Please re-send the rendered images or the PDF pages 9 to 16 of MATH8-Q3-MODULE8.pdf. I’ll transcribe them with page markers 9 to 16, following the same structure and rules as before.

I know that (1)____________ is the triangle congruence postulate that can be used to prove that ∆ XYW ≅ ∆XYZ. The corresponding congruent sides are (2). The corresponding congruent angles are (3). If I put together the two triangles in such a way that XY̅ of ∆XYW coincides with XY̅ of (4), (5) is formed. The common side shared by the two triangles is (6). The adjacent angles formed when the common side divides ∠WXZ are (7). These adjacent angles are congruent because (8). XY̅ (9) ∠WXZ, hence it is an angle bisector. The adjacent angles formed when the common side intersects WZ̅ are (10)____________ which are both (11)____________ in measure. Because of this fact, it is rightful to call XY̅ as ____________.

What I Can Do

Situation: Your barangay will hold a kite-making contest and submission of kite design is required. On a clean bond paper, draw the design of the kite that you want as an entry for the contest. The following rubric will be used to judge your kite design.

Rubrics:

Categories and Criteria	4 Good	3 Fair	2 Poor
Design	Geometry, Right Angles, and Symmetry are properly illustrated in the design.	Any two of geometry, right angles and symmetry are illustrated in the design.	None or only one of geometry, right angles, and symmetry is illustrated in the design.
Creativity	Design incorporates artistic elements, 100% functional, and 100% original.	Design is artistic, 50% functional, and 50% original.	Design is basic, 20% functional, and 20% original.

Assessment

Post-Assessment

Directions: Read and answer each of the following items accurately. Write the letter of the correct answer on your answer sheet of paper.

1. Which of the following is true about an angle bisector?

A. It divides an angle into halves. B. It divides an angle at 45⁰. C. It divides an angle at 90⁰. D. It divides an angle into three parts.

2. What do you call the segments or lines that intersect forming a 90-degree angle?

A. angle bisector C. skew segments/lines B. auxiliary lines D. perpendicular segments/lines

For items 3 – 7, use the figure at the right to answer the questions that follow.

[illustration: triangles EHO and HP; segment HP drawn from H to base EO, with a right-angle mark at H and tick marks and an arc at P; labels E, H, O, P]

3. Which among the following theorems of triangle congruence can be used to prove that ∆HEP ≅ ∆HOP?

A. AAS B. HyA C. LA D. LL

4. What is the corresponding side of OP̅?

A. EH̅ B. EP̅ C. HO̅ D. HP̅

5. Which of the following is true about PH̅?

A. PH̅ bisects ∠EOP C. PH̅ bisects ∠PEO B. PH̅ bisects ∠EPO D. PH̅ bisects ∠POE

6. Which of the following reasons justifies PH̅ ≅ PH̅?

A. Definition of congruent segments B. Definition of Perpendicular C. Reflexive Property D. Congruent Parts of Congruent Triangles are Congruent

7. What other information must be added in the figure above to prove ∆HEP ≅ ∆HOP by SAS Postulate?

A. EH̅ ≅ EP̅ C. OH̅ ≅ EP̅ B. EP̅ ≅ OP̅ D. OH̅ ≅ HP̅

For items 8 – 11, use the figure at the right to answer the questions that follow.

[illustration: two triangles side by side; left triangle with vertices F, E, G and right triangle with vertices F, H, G; tick marks on sides; labels 1, 2, 3, 4, 5, 6 at the angles and sides]

8. When two triangles are put together in such a way that a pair of corresponding sides coincide, what figure is formed?

A. trapezoid C. square B. hexagon D. triangle

9. In the resulting figure in item 8, what is the common side of the ∆EFG and ∆HFG?

A. EF̅ B. EG̅ C. FG̅ D. FH̅

10. What postulate or theorem proves that ∆FEG ≅ ∆FHG?

A. AAS B. SAS C. SSS D. ASA

11. What is the relationship of ∠1 and ∠2?

A. They are congruent angles. B. They are linear pair angles. C. They are non-congruent angles D. They are supplementary angles.

For items 12-15, use the figure at the right to answer the questions that follow.

[illustration: triangles GED and GEF sharing side GE; labels D, E, F, G and numbers 1 to 6 at the angles and sides; tick marks and a right-angle mark at E]

12. What postulate or theorem illustrates that ∆GED ≅ ∆GEF?

A. ASA C. HyL B. HyA D. LL

13. If you put together the two triangles in such a way that GE̅ of ∆GED coincides with the GE̅ of ∆GEF, what are the adjacent angles formed when the common side divides ∠DGF?

A. ∠1 and ∠2, ∠3 and ∠6 C. ∠1 and ∠4, ∠2 and ∠5 B. ∠1 and ∠2, ∠2 and ∠5 D. ∠1 and ∠4, ∠3 and ∠6

14. If you put together the two triangles in such a way that GE̅ of ∆GED coincides with the GE̅ of ∆GEF, which of the following is NOT TRUE about GE̅ to ∠DGF?

A. GE̅ bisects ∠DGF C. GE̅ is in the interior of ∠DGF B. GE̅ is in the exterior of ∠DGF D. GE̅ is the angle bisector of ∠DGF

15. If you put together the two triangles in such a way that GE̅ of ∆GED coincides with the GE̅ of ∆GEF, what is the relationship of GE̅ and DF̅?

A. GE̅ bisects DF̅ C. GE̅ do not lie in side DF̅ B. GE̅ is congruent to DF̅ D. GE̅ is parallel to DF̅

Additional Activities

Consider the simple house drawing with parts properly labeled and answer the questions that follow.

[illustration: simple house drawing with labeled points A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q; a triangular roof with apex A and base PG, a chimney BCDE, a vertical line AQ, a square LMIJ with diagonal MJ and a black dot at K, and a rectangle NOFH below the base]

Questions:

1. Using a protractor, which parts of the house drawing are perpendicular segments?
2. How did you determine that the segments are perpendicular?
3. Using a protractor, which parts of the house are angle bisectors?
4. How did you determine that the segments are angle bisectors?
5. How important are perpendicular lines and angle bisectors in building a house?

References

Abuzo,E.P., Bryant,M.L., Cabrella,J.B., Caldez,B.P., Callanta,M.M., Castro,A.I., Halabaso,A.R., et.al (2013). Mathematics 8 – Learner’s Module. Department of Education, Pasig City, Philippines, pp. 364-366.

Baccay, A. Geometry for Secondary Schools. Philippines: Phoenix Publishing House), pp. 140-190

Manalo,C.B, Suzara J.L., Mercado J.P.,Esparraga M.S.,Jr.Reyes N.V.,New Century Mathematics. Phoenix Publishing House

Ulpina,J.N., Tizon,L.T.,(2014). Math Builders 8. JO-ES Publishing House, Inc., pp.318-322.

Answer Key

What I Know

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

What’s In

1. SSS Postulate
2. ASA Postulate
3. Hypotenuse Acute Theorem
4. Hypotenuse-Leg Theorem
5. AAS Theorem
6. Leg-Acute Theorem
7. SAS Postulate
8. Leg-Leg Theorem

Guide Questions:

1. Through the use of markings.
2. Two triangles are congruent if the corresponding parts are congruent.

What’s New

1. The structure of the kite will not be balanced.
2. Yes
3. The left and right part of the kite must be equal.
4. The sides containing the upper part of the kite should be equal. The sides containing the lower part of the kite should be equal.
5. To have a good kite, it must have two pairs of equal length-sides that are adjacent to each other. Its diagonals must form a 90-degree angle.

What’s More

Activity: Answer Me!

A.

1. SSS Congruence
2. HJ ≅ HK, JI ≅ KI, and HI ≅ HI
3. ∠4 ≅ ∠1, ∠5 ≅ ∠2, and ∠6 ≅ ∠3
4. A kite/ kite HJIK
5. Yes because it is the common side.
6. ∠5 ≅ ∠2 and ∠4 ≅ ∠1
7. The pairs of adjacent angles are congruent as they are corresponding parts of congruent triangles.
8. HI bisects both ∠JHK and ∠JIK, hence HI is an angle bisector.

B.

1. LL Congruence Theorem
2. MN ≅ MN, NO ≅ NP
3. ∠4 ≅ ∠1, ∠5 ≅ ∠2, and ∠6 ≅ ∠3
4. A triangle/ ΔMPO
5. MN
6. ∠5 ≅ ∠2, ∠4 ≅ ∠1
7. Yes, because they are corresponding parts of the congruent triangles.
8. MN bisects ∠OMP, hence it is an angle bisector.
9. ∠MNP ≅ ∠MNO or ∠5 ≅ ∠2
10. Yes, because they are both right angles.
11. OP ⊥ MN

What I Have Learned

I know that LL Theorem is the triangle congruence postulate that can be used to prove ΔXYZ ≅ ΔXYW. The congruent corresponding sides are XW and XZ, WY and ZY, and XY and XY. The congruent corresponding angles are ∠1 and ∠4, ∠2 and ∠5, and ∠3 and ∠6. If I put together the two triangles in such a way that XY of ΔXYW coincides with XY of ΔXYZ, a kite is formed. The common side shared by the two triangles is XY. The adjacent angles formed when the common side divides ∠WXZ are ∠1 and ∠4 and ∠3 and ∠6. These adjacent angles are congruent because they are corresponding parts of the congruent triangles. XY bisects ∠WXZ, hence it is an angle bisector. The adjacent angles formed when the common side intersects WZ are ∠XYW and ∠XYZ which are both 90 degrees in measure. Because of this fact, it is rightful to call XY as perpendicular bisector.

Assessment

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

Additional Activities

1. AQ and PG, EB and BC, CD and BC, PO and NO, FG and FH, NH and NO, FH
2. By the use of protractor and look for segment that form 90 degrees.
3. MJ and AQ
4. By the use of protractor and look for segments that divide angles into two equal parts.
5. (Answers will vary with students)

For inquiries or feedback, please write or call:

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

Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600

Telefax: (632) 8634-1072; 8634-1054; 8631-4985

Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph