Republic of the Philippines

Department of Education Regional Office IX, Zamboanga Peninsula

Mathematics Quarter 3 - Module 6:

Proves Two Triangles are Congruent

Zest for Progress

Zeal of Partnership

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Name of Learner: ___________________________

Grade & Section: ___________________________

Name of School: ___________________________

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What I Need to Know

In this module, you will learn to:

a) define SSS, SAS, ASA, AAS and HL postulate; and

b) use the two-column to prove triangles are congruent.

What I Know

Directions: Write only the letter of the correct answer on the given space. For numbers 1-5, complete the Two-column proof below. G

Given: EDG GEF, DEG and GEF are right angles.

Prove: EF EG

Statements Reasons

EDG F Given

1. Given

GE GE 2.

3. All right angles are

DGE 4.

EF ED 5.

____1. a. DEG GEF b. EF EG c. DEG DEF

____2. a. Given b. ASA Postulate c. Reflexive Property

____3. a. GED and GEF b. EF EG c. DEG GEF

____4. a. ASA Postulate b. SAS Postulate c. CPCTC

____5. a. ASA Postulate b. SAS Postulate c. CPCTC

D F E

2

For numbers 6-10, complete the two-column proof below:

Given : XY ZW

YZ WX

Prove : WX YZ

Proof:

Statements Reasons

6. Given

YZ WX 7.

8. Reflexive Property

XYZ ZWX 9.

10. If alternate interior angles are congruent, lines are parallel

_____6. a. XY ZW b. XZ XZ c. XYZ ZXW

_____7. a. Given b. Reflexive Property c. SSS Postulate

_____8. a. XY ZW b. XZ XZ c. XYZ ZXW

_____9. a. Reflexive Property b. CPCTC c. SSS Postulate

_____10. a. XY ZW b. WX YZ c. XYZ ZXW

What’s In

Activity1: “Correct me if I’m wrong”

Directions: Read the statements carefully. Put a check mark on the space provided if the underlined word/phrase in the statement is true, otherwise write the correct

word/phrase that will make the it true.

1. The SSS Postulate (________) states that if three sides of one triangle are congruent to three sides of another triangle, then these two triangles are congruent.

2. Congruent (____________) triangles are triangles with equal measurements in three sides and three angles.

3. The HL Postulate states that if the hypotenuse and leg of any (________) triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are

congruent. 4. SSS, SAS, ASA, AAS and AAA (________) Postulates are five ways to prove that two

triangles are congruent. 5. The converse of CPCTC is that every corresponding part of two triangles are

congruent, then the triangles are congruent (___________).

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What’s New

Activity 2: “Do you know us?” Directions: Read and understand each statement carefully. Match Column A with

Column B. Column A Column B

1. I am a polygon with three sides and three angles. a. HL Postulate 2. The two of us are equal. b. Triangle 3. They use these to prove that we are equal. c. congruent

4. They used this proof when we are right angle triangles d. Postulate 5. We can be proven using our three sides. e. ASA Postulate

f. SSS Postulate

What is it

Two triangles are congruent if they have:

exactly the same three sides and exactly the same three angles.

We don't have to know all three sides and all three angles. Knowing three out of the six is enough.

There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

1.) SSS stands for "side, side, side" and means that we have two triangles with all three

sides equal.

For example:

is congruent to

SSS Congruence Postulate

If three sides of a triangle are congruent respectively to three

sides of another triangle, then the two triangles are congruent.

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Given: E is the midpoint of AC and BD

AB DC

Prove: AEB CED

Proof:

Statement Reason

1. E is the midpoint of AC and BD 1.Given

2. AE CE, BE DE 2.Definition of a midpoint

3. AB DC 3.Given

4. AEB CED 4.SSS Congruence Postulate

2.) SAS stands for "side, angle, side" and means that we have two triangles where we

know two sides and the included angle are equal.

For example:

is congruent to

Given: DC AB at its midpoint C.

Prove: DA DB

Proof:

Statement Reason

1. The midpoint of AB is C 1.Given

2. AC CB 2.Definition of a Midpoint

3. DC DC 3.Reflexive Property

4. DCA and DCB are right angles

4.Definition of Perpendicular lines

5. DCA DCB 5.All rights angles are congruent

6. DCA DCB 6.SAS Congruence Postulate

7. DA DB 7.CPCTC

In the example above, DC is called the perpendicular bisector of AB.

SAS Congruence Postulate

If two sides and the included angle of one triangle are congruent

respectively to two sides and the included angle of another triangle, then

the two triangles are congruent.

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3.) ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.

For example:

is congruent to

Given: E is the midpoint of AD,

EA bisects BEF, A D

Prove: ABE DCE

Proof:

Statement Reason

1. E is the midpoint of AD 1.Given

2. AE DE 2.Definition of a Midpoint

3. EA bisects BOP 3.Given

4. 4.Definition of an angle bisector

5. 5.Vertical Angle Theorem

6. 6.Transitivity

7. A D 7.Given

8. ABE DCE 8.ASA Congruence Postulate

4.) AAS stands for "angle, angle, side" and means that we have two triangles where we

know two angles and the non-included side are equal.

For example:

is congruent to

ASA Congruence Postulate

If two angles and the included side of one triangle are congruent

respectively to two angles and the included side of another triangle, then

the two triangles are congruent.

AAS Congruence Postulate

If two angles and a non-included side of one triangle are congruent

to two angles and non-included side of another triangle, then the two

triangles are congruent.

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Given: D and B are right angles.

Prove: ABC CDA

Statement Reason

1. D and B are right angles 1.Given

2.

3. D B 2.All right angles are congruent

4. 3.Given

5. AC AC 4.Reflexive Property

6. ABC CDA 5.AAS Congruence Postulate

5.) HL stands for "Hypotenuse, Leg" (the longest side of a right-angled triangle is called the "hypotenuse", the other two sides are called "legs")

It means we have two right-angled triangles with

the same length of hypotenuse and the same length for one of the other two legs.

It doesn't matter which leg since the triangles could be rotated.

This one applies only to right angled-triangles!

For example:

is congruent to

HyL Congruence Theorem

If a leg and the hypotenuse of one right triangle are congruent to a

corresponding leg and the hypotenuse of another right triangle, then the

triangles are congruent.

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Example Given: CD EA, AD is the perpendicular

Bisector of CE.

Prove: CBD EBA

Proof:

Statements Reasons

1. AD is the bisector of CE 1.Given

2. CBD EBA are right 2.Definition of Perpendicular Lines

3. B is the midpoint of CE 3.Definition of Bisector

4. CBD EBA are right angles 4.Definition of right angles

5. CB JEB 5.Definition of Midpoint

6. CD EA 6.Given

7. CBD EBA 7.HyL Congruence Theorem

What’s More

Activity 3: “Constant practice makes one perfect” Directions: Prove that the triangles are congruent. Use the two-column proof.

1. Given: AC ED

AB EF

BC FD

Prove: ABC EFD

STATEMENT REASON

2. Given : X is the midpoint of VZ

1 2

Prove : VXW ZXY

STATEMENT REASON

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What I Have Learned

Activity 4: “Fill me up” Directions: Prove that the triangles are congruent. Use the two-column proof.

1. Given :

HG FG

Prove : EHG EFG

STATEMENT REASON

2. Given:

Prove: EHG FHG

STATEMENT REASON

What I Can Do

Activity 5: “Your talent, your gift”

Directions: Choose among the five postulates to prove that two neckerchiefs are congruent. The rubric for your activity is shown below.

CRITERIA Outstanding

4 Satisfactory

3 Developing

2 Beginning

1 Rating

Accuracy

The computations are accurate and show wise use of the concepts of triangle congruence.

The computations are accurate and show use of the concepts of triangle congruence.

Some computations are erroneous and show use of some concepts of triangle congruence.

The computations are erroneous and do not show use of the concepts of triangle congruence.

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Creativity

The design is comprehensive and displays the aesthetic aspects of the mathematical concepts learned.

The design is presentable and makes use of the concepts of geometric representation.

The design makes use of the geometric representations but not presentable.

The design doesn’t use geometric representations and not presentable.

Stability

The design is stable and comprehensive and displays the aesthetic aspect of the principles of triangle congruence.

The design is stable, presentable and makes use of congruent triangle.

The design makes use of triangles, but not stable.

The design does not use triangles and is not stable.

Mathematical

Reasoning

The explanation is clear, exhaustive or thorough, and coherent. It includes intersecting facts and principles.

The explanation is clear and coherent. It covers the important concepts.

The explanation is understandable but not logical.

The explanation is incomplete and inconsistent.

OVERALL RATING

Assessment Directions: Choose the letter of the best answer. Write the letter chosen on the space

provided for you.

For numbers 1-5, Complete the Two-column proof below

Given : XY ZW

YZ WX

Prove : WX YZ

Proof:

Statements Reasons

1. Given

YZ WX 2.

3. Reflexive Property

XYZ ZXW 4.

5. If alternate interior angles are congruent, lines are parallel.

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_____1. a. XY ZW b. XZ XZ c. XYZ ZXW

_____2. a. Given b. Reflexive Property c. SSS Postulate

_____3. a. XY ZW b. XZ XZ c. XYZ ZXW

_____4. a. Reflexive Property b. CPCTC c. SSS Postulate

_____5. a. XY ZW b. WX YZ c. XYZ ZXW

For numbers 6-8, Complete the two-column proof below.

Given: RN IA

R and I are right angles

Prove: RAN INA

Proof:

Statements Reasons

R and I are right angles Given

RAN and INA are right triangles

Definition of right triangles

6. Given

NA NA 7.

RAN INA 8.

____6. a. RN IA b. RA IN c.NA NA

____7. a. Transitive Property b. Reflexive Property c. Symmetric Property

____8. HyA Congruence b. CPCTC c. HyL Congruence

____9. What additional information do you need in order to prove that ABC is congruent

to DEF using HyL Congruence Theorem?

a. AC DF

b. AB DE

c. A D

____10. What additional information do you need in order to prove that ACD is

congruent to DBA using the HL Congruence Theorem?

a. AB DC

b. BD CD

c. AD AD

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Math 8 Quarter 3 Module 6 KEY ANSWER

What I Know 1. a 6. a

2. c 7. a 3. a 8. b

4. a 9. c 5. c 10. b

What’s In Activity1: “Correct me if I’m wrong”

1. / 2. /

3. right 4. HL 5. /

What’s New

Activity 2 – “Do you know us?” 1. b

2. c 3. d 4. a

5. f

What’s more Activity 3 – “Constant practice makes one perfect” 1.

STATEMENT REASON

AC ED 1. Given

2. AB EF 2.Given

3. BC FD 3.Given

4. ABC EFD 4.SSS Congruence Postulate

2.

STATEMENT REASON

1.X is the midpoint of VZ

1.Given

2.VX ZX 2.Definition of Midpoint

3. ZXY 3.AAS Congruence

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What I have learned

Activity 4 – “Fill me up” 1.

Statements Reasons

1. 1 2 1.Given

2. HG FG 2.Given

3. GE GE 3.Reflexive Property

4. EHG EFG 4. SAS Congruence Postulate

2.

Statements Reasons

1. 1 2 1.Given

2. 3 4 2.Given

3. GH GH 3.Reflexive Property

4. EHG FHG 4. ASACongruence Postulate

What I can do

Activity 5 – “Your talent, your gift”

Output varies

Assessment 1. a

2. a 3. b

4. c 5. b 6. a

7. b 8. c

9. a 10. a

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References

Bass, L. et al. Geometry: Tools For A Changing World. Prentice-Hall, Inc., 1998,

406-439. Boyd, C. et al. Texas Geometry. McGraw-Hill Companies, Inc., 2008, 234-251.

Glorial, J. et al. 21st Century Mathematics: A Vision For A Better Future. Phoenix

Publishing House, Inc., 2013, 49-355.

Serra, M. Discovering Geometry: An Investigative Approach. Key Curriculum Press, 2003, 230-240.

Pierce, Rod. (12 Jun 2017). How To Find if Triangles are Congruent. Math Is Fun.

Retrieved 29 Nov 2020 from http://www.mathsisfun.com/geometry/triangles-congruent-finding.html

Jenn (2020). Congruent Triangles. Retrieved 29 Nov 2020 from https://calcworkshop.com/congruent-triangles/sss-sas-postulates/

Development Team

Writer: Geraldine E. Magallon

Monching National High School

Editor/QA: Eugenio E. Balasabas

Ressme M. Bulay-og Mary Jane I. Yeban

Reviewer: Gina I. Lihao

EPS-Mathematics

Illustrator:

Layout Artist:

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Jerry c. Bokingkito OIC-Assistant SDS

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