Mathematics 5 Quarter 3 Module 8: Visualizing Congruent Polygons

Mathematics Quarter 3 – Module 8: Visualizing Congruent Polygons

GOVERNMENT PROPERTY NOT FOR SALE [stamp]

CO_Q3_Mathematics 5_ Module 8

Copyright and Development Team

Mathematics – Grade 5 Alternative Delivery Mode Quarter 3 – Module 8: Visualizing Congruent Polygons First Edition, 2020

Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties.

Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.

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

Printed in the Philippines by ________________________

Development Team of the Module

Writers: Ma. Alma M. Galitan

Editors: Niann L. Atis

Reviewers: Renato S. Cagomoc, Rolando Lacbo, Joshua Sherwin T. Lim, Ramil A. Tan, Robert C. Guira

Illustrator: Noel E. Sagayap

Layout Artist: Razle L. Jabelo, Emmanuel S. Gimena Jr.

Management Team: Ma. Gemma M. Ledesma, Arnulfo R. Balane, Rosemarie M. Guino, Joy B. Bihag, Ryan R. Tiu, Sarah S. Cabaluna, Thelma Cabadsan-Quitalig, Elena S. De Luna, Renato S. Cagomoc, Noel E. Sagayap, Geraldine P. Sumbise, Joshua Sherwin T. Lim

Department of Education – Region VIII

Office Address: DepEd Regional Office No. 8, Candahug, Palo, Leyte Telefax: (053)-832- 2997 E-mail Address: region8@deped.gov.ph

Mathematics Quarter 3 – Module 8: Visualizing Congruent Polygons

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

[icon: target]

Good day Mathletes! The idea of equality in numbers is now extended to figures and this is referred to as congruence. You can see congruent figures all around us. Go to a grocery and what do you see? Packs of drinks are of uniform shapes and sizes. In this module, you are going to understand and visualize congruent polygons. You will learn what makes shapes or plane figures congruent.

After going through this module, you are expected to:

1. visualize congruent polygons.

Before you begin the lesson, first, let us check what you know about congruent polygons.

What I Know

[icon: pencil]

A. Directions: Write the letter of your choice of answer on a separate sheet of paper.

1. When are two figures congruent?

• A. They are both line segments.
• B. They have the same size.
• C. They have the same shape.
• D. They have the same size and shape.

2. [illustration: two triangles, one shown as a right triangle and the other as a right triangle reflected beside it]

Are these two triangles congruent?

• A. No
• B. Yes
• C. Maybe
• D. Never

3. In congruent figures, the corresponding sides are congruent, and the corresponding angles are ________________.

• A. congruent
• B. complementary
• C. non-congruent
• D. supplementary

4. Among the figures below, which one is most likely to be congruent to [box: reference square]?

• A. [illustration: diamond-shaped figure, a square rotated 45 degrees]
• B. [illustration: rectangle]
• C. [illustration: pentagon-like figure with a pointed right side]
• D. [illustration: rectangle, wider than tall]

5. Matching sides are called _______________ sides.

• A. equal
• B. opposite
• C. congruent
• D. corresponding

6. Which two-line segments are congruent in the figure?

• A. $\overline{AB}$ and $\overline{BC}$
• B. $\overline{AB}$ and $\overline{AC}$
• C. $\overline{BC}$ and $\overline{BD}$
• D. $\overline{AC}$ and $\overline{BD}$

[illustration: a line segment with points A, B, C, and D marked on it, drawn above a ruler marked 0 to 12 in mm]

7. [illustration: two figures placed side by side, the first a cross-like polygon and the second a rectangle-like polygon with notched sides]

Are these figures congruent?

• A. No
• B. Yes
• C. Maybe
• D. Never

8. How do you call this given pair of triangles?

• A. congruent
• B. corresponding
• C. non-congruent
• D. same triangle

9. Which of the following is TRUE for any two congruent polygons?

• A. Their areas are equal.
• B. Their perimeters are equal.
• C. They can be made to fit together when superimposed.
• D. All of the above

10. Which of the following is a pair of congruent polygons?

• A. Any two right triangles
• B. Any two isosceles triangles
• C. Any two rectangles
• D. None of these

Lesson 1 Visualizing Congruent Polygons

Two polygons are congruent if and only if their corresponding sides and angles are congruent. Thus, polygons that have the same size and shape are congruent. Success in completing this module shows that you have understood what polygons are, their types, their properties and when they are congruent. Learning about polygons does not just end in the identification of shapes, sizes and properties. Knowledge on visualizing congruent polygons has an important role in our daily living such as the awareness of its presence in nature, in art, in human-built structures and its other applications all around us.

Are you ready to learn? Let’s explore the lesson.

What’s In

[icon: puzzle pieces]

Before we go further on visualizing congruent polygons, let us review the previous lesson. Let’s check if you can still remember how to describe and compare properties of regular and irregular polygons.

Regular Polygon

• It is a polygon that is both equilateral and equiangular. Equilateral means all sides have equal measures. Equiangular means all angles have equal measures.
• A square is a regular polygon because all of its angles are right angles and all of its sides have equal measures.

Examples of regular polygon

• [illustration: equilateral triangle with each side marked equal and each angle labeled 60°]
• [illustration: regular pentagon with each side labeled 5 cm and each interior angle labeled 108°]
• [illustration: square with each side labeled 6 cm and right angles marked]

A “Regular Polygon” has:

• all sides equal and
• all angles equal.

Otherwise, it is irregular.

[illustration: regular pentagon, captioned “Regular Pentagon”] [illustration: irregular pentagon, captioned “Irregular Pentagon”]

Irregular Polygon

• It is a polygon that has sides with different measures and has angles with different measures.
• A rectangle is an irregular polygon because all of its angles are right angles, but not all of its sides have the same measures.

Examples of irregular polygon

[illustration: triangle with sides labeled 7 cm, 10 cm, and 9 cm] [illustration: right triangle with a right-angle mark] [illustration: rectangle with tick marks on its sides]

Polygon/Shape	Regular	Irregular
Triangle	[illustration: regular triangle]	[illustration: irregular triangle]
Quadrilateral	[illustration: square]	[illustration: irregular quadrilateral]
Pentagon	[illustration: regular pentagon]	[illustration: irregular pentagon]
Hexagon	[illustration: regular hexagon]	[illustration: irregular hexagon]
Heptagon	[illustration: regular heptagon]	[illustration: irregular heptagon]
Octagon	[illustration: regular octagon]	[illustration: irregular octagon]

What’s New

In the previous module, you were taught how to describe and compare the properties of polygons (regular and irregular polygons). This module is going to teach and guide you how to visualize congruent polygons.

Always remember that the word “congruent” is defined as “identical in form; coinciding exactly when superimposed.” The origin of the word “congruent” is from the Latin to “agree” or “to meet together”.

Congruent polygons are exactly the same size and exactly the same shape. All of their sides have the same length and all of their angles have the same measure.

Example 1:

Study the following figures. What can you say about each set of figures?

Set A:

[illustration: two hexagons with tick marks on their sides]

Set B:

[illustration: a parallelogram and a trapezoid]

Set C:

[photo: a wall clock with an octagonal frame] [illustration: a puzzle-like polygon with curved cutouts]

Set D:

[photo: an umbrella viewed from above, divided into sections] [photo: a 2-peso coin with a palm tree design]

What is It

In our everyday encounter with objects, structures, shapes and figures, we can observe polygons in them in different forms, colors and sizes. Understanding how to visualize congruent polygons is easy when you have also mastered the skill on visualizing, naming and describing polygons with 5 sides or more as well as on describing and comparing properties of regular and irregular polygons.

Polygons are congruent if they are equal in all respects: same number of sides; all corresponding sides have the same length; and all corresponding interior angles have the same measure.

Intuitively, congruent figures are of the same size and shape. It is possible to turn, flip and/or slide one figure so it will fit exactly on the other figure. The areas of congruent figures are equal. The perimeters of congruent figures are equal. The corresponding angles of congruent figures are equal in measure. The corresponding line segments of congruent figures are equal in length.

Congruent Figures

• These are pairs of figures which have exactly the same size and shape.
• These figures have equal measures for all the sides and angles.

Congruent Polygons

• Two polygons are congruent if their corresponding sides and angles are congruent.

Let’s consider the polygons shown above.

Congruent polygons

Set A:

[illustration: two congruent quadrilaterals with tick marks on their sides]

Not Congruent polygons

Set B:

[illustration: a parallelogram and a trapezoid]

Not Congruent polygons

Set C:

[photo: a wall clock with an octagonal frame] [illustration: a puzzle-like polygon with curved cutouts]

Not Congruent polygons

Set D:

[photo: an umbrella viewed from above, divided into sections] [photo: a 2-peso coin with a palm tree design]

Identify whether the following are congruent polygons.

[chart: two isosceles triangles; left triangle sides labeled 10 cm, 10 cm, 16 cm and angles 78°, 51°, 51°; right triangle sides labeled 20 cm, 20 cm, 32 cm and angles 78°, 51°, 51°]

The two triangles have equal measures of angles. However, notice that the measures of their sides are not equal. This means that the two triangles are not congruent.

[illustration: two triangles with tick marks on their sides, the second rotated]

The tick marks indicate that the corresponding sides of the two triangles are congruent. The second triangle is just a rotated version of the first triangle. Therefore, the triangles are congruent

What’s More

Activity 1: Find My Pair!

Directions: Identify the figures that appear to be congruent. Write your answer on a separate sheet of paper.

• A. [illustration: square rotated to a diamond]
• B. [illustration: triangle]
• C. [illustration: square rotated to a diamond]
• D. [illustration: circle]

• A. [illustration: parallelogram]
• B. [illustration: rectangle]
• C. [illustration: trapezoid]
• D. [illustration: rectangle]

• A. [illustration: hexagon]
• B. [illustration: hexagon]
• C. [illustration: pentagon]
• D. [illustration: rounded square]

• A. [illustration: arrow pointing down]
• B. [illustration: arrow pointing right]
• C. [illustration: arrow pointing up]
• D. [illustration: arrow pointing left]

• A. [illustration: triangle]
• B. [illustration: triangle pointing down]
• C. [illustration: triangle]
• D. [illustration: triangle pointing down]

Activity 2: Be True to Me!

Directions: Identify whether the statement is True or False. Write your answer on a separate sheet of paper.

1. Congruent polygons have the same shape and size.
2. The shape and size are not considered in congruent polygons.
3. The two congruent polygons must fit on one another.
4. In congruent polygons, their corresponding angles and sides are congruent.
5. The color of the polygons is not considered in deciding whether two polygons are congruent.

Activity 3: Yes or No!

Directions: For items 1 -4: Analyze the given polygons if they are congruent or not. Write Yes or No only for your answers. Write your answer on a separate sheet of paper.

[illustration: parallelogram ABCD with diagonal AC; tick marks on sides]

[illustration: two small triangles with tick marks; one labeled with vertices A, B, C and a side of 4.4 cm, the other with vertices X, Y, Z]

[illustration: triangle RST with sides 72 mm and 72 mm and angle marks; triangle LMN with sides 75 mm and 75 mm and angle marks]

[illustration: two triangles with vertices E, F, G, H, A, B, C, D; side lengths 4.2, 3, 1.4, and 4.1 labeled]

5. Are two polygons congruent if their corresponding sides have the same length?

What I Have Learned

Directions: Fill in the blanks with the correct word or group of words to make the statement complete. Write your answer on a separate sheet of paper.

(1)_____ are congruent if, (2)_____ have the same (3)_____ and (4). Tracing one perfectly (5) the other, their corresponding (6)_____ and sides are (7). All corresponding sides have the (8) length; and all corresponding (9)_____ have the same (10)_____.

Good job! Just keep going.

What I Can Do

Directions: Match the polygons in both columns that are most likely to be congruent. Write your answer on a separate sheet of paper.

Column A	Column B
1. [illustration: tall rectangle]	a. [illustration: triangle pointing down]
2. [illustration: cross-shaped polygon]	b. [illustration: pentagon]
3. [illustration: trapezoid]	c. [illustration: trapezoid]
4. [illustration: triangle]	d. [illustration: rectangle]
5. [illustration: pentagon]	e. [illustration: hexagon]
	f. [illustration: cross-shaped polygon]

Assessment

A. Directions: Choose the figure that is likely to be congruent to the given figure. Write your answer on a separate sheet of paper.

1. [illustration: shaded triangle]

• A. [illustration: small triangle]
• B. [illustration: triangle]
• C. [illustration: tall triangle]
• D. [illustration: long slanted triangle]

2. [illustration: shaded hexagon]

• A. [illustration: tall hexagon]
• B. [illustration: flattened hexagon]
• C. [illustration: wide hexagon]
• D. [illustration: small hexagon]

3. [illustration: shaded trapezoid]

• A. [illustration: parallelogram]
• B. [illustration: trapezoid]
• C. [illustration: tall trapezoid]
• D. [illustration: rectangle]

4. [illustration: shaded right triangle]

• A. [illustration: tall right triangle]
• B. [illustration: narrow right triangle]
• C. [illustration: small right triangle]
• D. [illustration: triangle]

5. [illustration: shaded semicircle-topped shape]

• A. [illustration: shape with rounded top and flat bottom]
• B. [illustration: wide rounded shape]
• C. [illustration: arched shape]
• D. [illustration: pentagon-like shape]

B. Directions: Construct the pair of polygons being described. Write your answer on a separate sheet of paper.

1. equiangular triangles

[illustration: empty box for answer]

2. rhombuses with 15-decimeter sides

[illustration: empty box for answer]

3. right triangles with side lengths 6 cm, 8 cm, and 10 cm.

   
   

4. isosceles triangles – in each triangle, the congruent sides measure 12 cm each.

   
   

5. squares with 20-centimeter sides

   
   

[transcriber’s note: three empty answer boxes are printed beside items 3 to 5]

Answer Key

[transcriber’s note: the three answer-key boxes are printed upside down on this page. They are transcribed in the order they read when the page is turned 180 degrees.]

What’s More

Activity 2

6. True
7. False
8. True
9. True
10. True

Activity 1

1. A & C
2. B & D
3. A & B
4. A & C
5. A & D

What I Know

1. D
2. B
3. A
4. A
5. D
6. D
7. A
8. A
9. D
10. D

What I Can Do

1. D
2. F
3. C
4. A
5. B

What I Have Learned

1. Polygons
2. Both/They
3. Shape
4. Size
5. Fits
6. Angles
7. Congruent
8. Same
9. Interior
10. Measure

What’s More

Activity 3

1. Yes
2. Yes
3. No
4. No
5. No

Assessment

A.

1. B
2. D
3. B
4. A
5. A

B.

1. [illustration: two gray triangles with side labels, not legible]
2. [illustration: two gray parallelograms with side labels, partly legible: 15 dm, 12 dm [?]]
3. [illustration: right triangles with labels 6 cm, 8 cm, and 10 cm]
4. [illustration: two isosceles triangles, each with congruent sides labeled 12 cm]
5. [illustration: two squares, each labeled 20 cm]

References

“Congruent Polygons: Definition & Examples.” Study.com. Accessed March 3, 2021. https://study.com/academy/lesson/congruent-polygons-definition-examples.html.

Lumbre, U., Placer, B., & Sy, Jr., S. (2016). 21st Century Mathletes Textbook. Vibal Group Inc. s

Donna Roberts and , Frederick Roberts. Concept of Congruence – MathBitsNotebook(Geo – CCSS Math). Accessed March 3, 2021. https://mathbitsnotebook.com/Geometry/CongruentTriangles/CTConcept.html.

Foundation, CK-12. “12 Foundation.” CK. Accessed March 3, 2021. https://www.ck12.org/book/ck-12-middle-school-math-concepts-grade-8/section/6.10/.

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