Mathematics 8 Quarter 3 Module 7: Proving Statements on Triangle Congruence

Mathematics Quarter 3 – Module 7 Proving Statements on Triangle Congruence

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Mathematics – Grade 8 Alternative Delivery Mode Quarter 3 – Module 7 Proving Statements on Triangle Congruence First Edition, 2021

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Mathematics Quarter 3 – Module 7 Proving Statements on Triangle Congruence

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 is designed for you to understand how to prove statements on triangle congruence. You will be able to know how to prove that the corresponding parts of congruent triangles are congruent. The scope of this module enables you to use it in many different learning situations. The lesson is arranged to follow the standard sequence of the lesson. 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: Proving Statements on Triangle Congruence

After going through this module, you are expected to:

1. identify statements on triangle congruence;
2. apply the postulates and theorems on triangle congruence to prove statements involving (a) multiple angles, (b) isosceles triangle, (c) overlapping triangles; and
3. relate the importance of proving statements on triangle congruence in real life situations.

What I Know

Pre-Assessment

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

1. Which of the following refers to the triangles that have common region?
   A. Multiple triangles
   B. Isosceles triangles
   C. Congruent triangles
   D. Overlapping triangles

   
   

2. Which of the following theorems states that, “if two sides of a triangle are congruent, then the angles opposite them are congruent.”?
   A. Right Angle Theorem
   B. Isosceles Triangle Theorem
   C. Interior Angles of a Triangle Theorem
   D. Exterior Angles of a Triangle Theorem

   
   

For items 3 and 4, use the figure at the right. [illustration: triangle ABC with vertices A (bottom left), B (top), C (bottom right), and a point D on side AB joined to C by a segment; labels A, B, C, D as printed]

3. In an isosceles $\triangle ABC$, let $\overline{CD}$ be an angle bisector of $\angle BCA$. What theorem or postulate can justify $\triangle DCA \cong \triangle DCB$?
   A. Angle-Angle-Side
   B. Hypotenuse-Acute Angle
   C. Hypotenuse-Leg
   D. Side-Angle-Side

   
   

4. In item number 3, $\triangle DCA \cong \triangle DCB$. What other corresponding sides are congruent by CPCTC aside from $\overline{CD} \cong \overline{CD}$?
   A. $\overline{CD} \cong \overline{CA}$
   B. $\overline{CD} \cong \overline{CB}$
   C. $\overline{DA} \cong \overline{DB}$
   D. $\overline{DA} \cong \overline{AC}$

   
   

For items 5 to 7, use the figure below. [illustration: triangle ABC with vertices B (top), A (bottom left), C (bottom right) and a segment BD drawn to point D on side AC; beside it, triangle EFG with vertices E (top), F (bottom left), G (bottom right); labels as printed]

5. Which of the following properties justifies the statement “$\overline{BD} \cong \overline{BD}$”?
   A. Equivalence
   B. Reflexive
   C. Symmetric
   D. Transitive

   
   

6. Which of the following properties justifies the statement “If $\overline{AD} \cong \overline{DC}$ and $\overline{DC} \cong \overline{FG}$, then $\overline{AD} \cong \overline{FG}$”?
   A. Equivalence
   B. Reflexive
   C. Symmetric
   D. Transitive

   
   

7. If $\overline{AB} \cong \overline{BC}$, which angles are guaranteed congruent by the Isosceles Triangle Theorem?
   A. $\angle ABD$ and $\angle CBD$
   B. $\angle BAD$ and $\angle BCD$
   C. $\angle EFG$ and $\angle BDC$
   D. $\angle FEG$ and $\angle EGF$

   
   

8. In the figure, $\triangle CBA \cong \triangle ADC$. Which of the following can justify the statement $\angle B \cong \angle D$?
   [illustration: quadrilateral with vertices A (bottom left), B (top left), C (top right), and D (bottom right), with diagonal AC drawn and tick marks on the sides]
   A. CPCTC
   B. Reflexive Property
   C. Definition of Midpoint
   D. Isosceles Triangle Theorem

   
   

9. The figure at the right are overlapping triangles where $\angle SIH \cong \angle NHI$ and $\overline{SI} \cong \overline{NH}$. Which of the following relations is true?
   [illustration: overlapping triangles with vertices H (top left), I (top right), S (bottom left), and N (bottom right), crossing at point E]
   A. $\triangle SHI \cong \triangle HIN$
   B. $\triangle SHI \cong \triangle HNI$
   C. $\triangle SHI \cong \triangle NHI$
   D. $\triangle SHI \cong \triangle NIH$

   
   

10. In $\triangle GEM$, let $\overline{ES}$ be a perpendicular bisector of $\overline{GM}$. If $\triangle GES \cong \triangle MES$, which side of the triangles is congruent by CPCTC?
    A. $\overline{ES} \cong \overline{EM}$
    B. $\overline{ES} \cong \overline{ES}$
    C. $\overline{GS} \cong \overline{ES}$
    D. $\overline{GE} \cong \overline{ME}$

    
    

For items 11 to 15. Complete the proof by filling in the blanks with the correct statements and reasons. Options are found in the box.

A. CPCTC B. Definition of Vertical Angles C. Given D. SAS Congruence Postulate E. $\triangle AGN \cong \triangle EGL$ F. Vertical Angle Theorem

[illustration: segments AE and NL crossing at point G, with labels A, G, E, L, N]

Given: $\overline{GA} \cong \overline{GE}$; $\overline{GN} \cong \overline{GL}$

Prove: $\overline{AN} \cong \overline{EL}$

Proof:

Statements	Reasons
1. $\overline{GA} \cong \overline{GE}$	1. Given
2. $\overline{GN} \cong \overline{GL}$	2. (11)
3. $\angle AGN$ and $\angle EGL$ are vertical angles.	3. (12)
4. $\angle AGN \cong \angle EGL$	4. (13)
5. (14)	5. SAS Congruence Postulate
6. $\overline{AN} \cong \overline{EL}$	6. (15)

What is It

Triangles are classified according to sides and angles. Some triangles have characteristics that come from both of these classifications like scalene right triangle and isosceles right triangle. Figure 1 is an example of scalene right triangle. Aside from having a right angle, no two sides are congruent while isosceles right triangle as shown in Figure 2 has congruent legs. First, let us review the parts of the scalene right and isosceles right triangle.

[illustration: right triangle ABC with vertices A (top), B (bottom left, right-angle mark), C (bottom right); labels “Acute angle” at A, “Hypotenuse” along side AC, “Acute angle” at C, “Leg” along side AB, “Leg” along side BC; callout text “ΔABC is a scalene right triangle because the legs $\overline{AB}$ and $\overline{BC}$ are not congruent.”]

Figure 1. Triangle ABC is a scalene right triangle.

[illustration: isosceles right triangle DEF with vertices D (bottom left), E (top, right-angle mark), F (bottom right); labels “Leg” along side DE, “Vertex Angle” at E, “Leg” along side EF, “Base” along side DF, “Base Angle” at D, “Base Angle” at F; callout text “ΔDEF is an isosceles right triangle because the legs $\overline{ED}$ and $\overline{EF}$ are congruent.”]

Figure 2. Triangle DEF is an isosceles right triangle.

The table shows the use of corresponding parts of scalene right triangle and isosceles right triangle in proving that two triangles are congruent. After proving that the two triangles are congruent, we can also say that the other corresponding parts of the congruent triangles are congruent (CPCTC). Here are some examples.

Congruent Triangles	Description	Other corresponding parts of congruent triangles that can be justified as congruent by CPCTC

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

Congruent Triangles	Description	Other corresponding parts of congruent triangles that can be justified as congruent by CPCTC
1. [illustration: two scalene right triangles, the first with vertices B, A, C and the second with vertices E, D, F, with tick marks on the sides]	The two scalene right triangles, ∆ABC ≅ ∆DEF by LL Congruence or HyL Congruence based on the marks in the figure. LL Congruence: $\overline{AB}$ ≅ $\overline{DE}$ $\overline{AC}$ ≅ $\overline{DF}$ HyL Congruence: $\overline{BC}$ ≅ $\overline{EF}$ $\overline{AB}$ ≅ $\overline{DE}$ or $\overline{BC}$ ≅ $\overline{EF}$ $\overline{AC}$ ≅ $\overline{DF}$	∠B ≅ ∠E ∠C ≅ ∠F $\overline{BC}$ ≅ $\overline{EF}$ ∠B ≅ ∠E ∠C ≅ ∠F $\overline{AC}$ ≅ $\overline{DF}$ ∠B ≅ ∠E ∠C ≅ ∠F $\overline{AB}$ ≅ $\overline{DE}$
2. [illustration: two scalene right triangles, the first with vertices H, G, I and the second with vertices K, J, L, with tick marks and angle marks]	The two scalene right triangles, ∆GHI ≅ ∆JKL by LA Congruence based on the figure. LA Congruence: $\overline{GH}$ ≅ $\overline{JK}$ ∠I ≅ ∠L	∠H ≅ ∠K $\overline{HI}$ ≅ $\overline{KL}$ $\overline{GI}$ ≅ $\overline{JL}$
3. [illustration: two isosceles right triangles, the first with vertices O, M, P and the second with vertices R, Q, S, with tick marks and angle marks]	The two triangles ∆MOP and ∆QRS are isosceles right triangle. ∆MOP ≅ ∆QRS by HyA Congruence Theorem. HyA Congruence: $\overline{OP}$ ≅ $\overline{RS}$ ∠P ≅ ∠S	∠O ≅ ∠R $\overline{MO}$ ≅ $\overline{RQ}$ $\overline{MP}$ ≅ $\overline{QS}$

The pairs of triangles shown in the table are congruent. However, not all problems involving congruent triangles looked like the examples above. We also have overlapping triangles that can also be proven congruent. Other problems may be presented like the figures below.

[illustration: Figure 3, a triangle with two segments crossing inside it, forming overlapping triangles]

[illustration: Figure 4, two overlapping triangles sharing a common base region]

[illustration: Figure 5, a rectangle with both diagonals drawn and right-angle marks at the two bottom corners]

Let us try to prove the congruence of overlapping triangles.

Illustrative Example 1

∆DAB and ∆CBA are overlapping scalene right triangles. In this example, we will prove that the other corresponding parts of congruent triangles are also congruent.

Direction: Complete the proof below by answering the guide questions provided in each missing statements and reasons.

1. Consider the given overlapping scalene right triangles.

Given: $\overline{DA}$ ⊥ $\overline{AB}$; $\overline{CB}$ ⊥ $\overline{AB}$; $\overline{DB}$ ≅ $\overline{CA}$

Prove: ∠ADB ≅ ∠BCA

[illustration: overlapping right triangles DAB and CBA inside a rectangle with vertices D (top left), C (top right), A (bottom left), B (bottom right), both diagonals drawn, right-angle marks at A and B]

Proof:

Statements	Reasons
1. ___________________ (What is the first Given statement?	1. Given
2. ___________________ (If $\overline{DA}$ ⊥ $\overline{AB}$, then what kind of angle is formed? What is the name of the angle formed?)	2. Definition of Perpendicular Lines
3. $\overline{CB}$ ⊥ $\overline{AB}$	3. Given
4. ∠CBA is a right angle	4. ___________________ (Based on the3rd statement, what do you think is the reason for the 4th statement?)
5. ___________________ (In the 2nd and 4th statements, formulate a statement based on the right angle theorem)	5. Right Angle Theorem
6. ___________________	6. Definition of right triangle

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

Statements	Reasons
(If ∠DAB and ∠CBA are right angles, what kind of triangle is ∆DAB and ∆CBA?)
7. $\overline{AB}$ ≅ $\overline{AB}$	7. Reflexive Property
8. ___________________ (What is the third given statement?	8. Given
9. ∆DAB ≅ ∆CBA	9. ___________________ (Based from statements 1 to 8, what theorem can be used to prove that the two triangles are congruent?)
10. ∠ADB ≅ ∠BCA	10.___________________ (After proving the two triangles are congruent, what statement can justify that the other corresponding parts are congruent?

Solution: (Complete Proof)

Statements	Reasons
1. $\overline{DA}$ ⊥ $\overline{AB}$	1. Given
2. ∠DAB is a right angle	2. Definition of Perpendicular Lines
3. $\overline{CB}$ ⊥ $\overline{AB}$	3. Given
4. ∠CBA is a right angle	4. Definition of Perpendicular Lines
5. ∠DAB ≅ ∠CBA	5. Right Angle Theorem
6. ∆DAB ≅ ∆CBA	6. Definition of right triangle
7. $\overline{AB}$ ≅ $\overline{AB}$	7. Reflexive Property
8. $\overline{DB}$ ≅ $\overline{CA}$	8. Given
9. ∆DAB ≅ ∆CBA	9. HyL Congruence Theorem
10. ∠ADB ≅ ∠BCA	10. CPCTC

In this example, we want to justify the congruence of ∠ADB ≅ ∠BCA. Based on the given information, which congruence theorem can be used? We are given two pair of perpendicular segments $\overline{DA}$ ⊥ $\overline{AB}$ and $\overline{CB}$ ⊥ $\overline{AB}$, which by definition, these pair of segments formed right angles, hence, the two triangles are right triangles. We are provided also with the given that $\overline{DB}$ ≅ $\overline{CA}$. Since the two triangles are overlapping, the two triangles shared the congruent segments by reflexive property. Thus, we can use this to justify the congruence of the corresponding hypotenuses and legs of the triangles. Now based on this, we can use the HyL congruence theorem to justify the congruence of the triangles.

The reason for numbers 1, 3 and 8 is given since it is a starting point or connector points in formulating other statements in proving triangle congruence. Once the congruence of two triangles is established, then, you can say that the other corresponding parts of congruent triangles are also congruent or also known as CPCTC. Thus, we can conclude that the last statement $\angle ADB \cong \angle BCA$ is proven by CPCTC.

Illustrative Example 2

Directions: Complete the proof below by considering the given clue in each missing statements and reasons.

Consider the figure at the right, $\triangle AVE$ is a right triangle and $\overline{VS} \perp \overline{AE}$. $S$ is the midpoint of $\overline{AE}$. Prove that $\triangle AVE$ is an isosceles triangle.

[illustration: right triangle with vertices E, V, A; point S on side EA; segment VS drawn from V to S; labels E, S, V, A]

Proof:

Statements	Reasons
1. $\overline{VS} \perp \overline{AE}$	1. Given
2. _____________________ (What kind of angles are formed when $\overline{VS} \perp \overline{AE}$)	2. Definition of Perpendicular lines
3. $\angle ASV \cong \angle ESV$	3. _____________________ (Theorem that states: All right angles are congruent)
4. ______________________________ (This statement is found in the given.)	4. Given
5. $\overline{ES} \cong \overline{AS}$	5. Definition of Midpoint
6. $\overline{VS} \cong \overline{VS}$	6. _________________ (What property of congruence that states any figure is congruent to itself?)
7. ____________________ (Name the two congruent triangles based on the previous statements.)	7. SAS Congruence Postulate
8. $\overline{AV} \cong \overline{EV}$	8. CPCTC
9. $\triangle AVE$ is an isosceles triangle	9. _____________________ (Based on the 9th statement, state the reason that justifies that $\triangle AVE$ is an isosceles triangle.)

Remember that when two rays, lines or line segments are perpendicular, there is a right angle formed. Moreover, when two legs of a right triangle are congruent, then the triangle is an isosceles right triangle.

Statements	Reasons
1. $\overline{VS} \perp \overline{AE}$	1. Given
2. $\angle ASV$ and $\angle ESV$ are right angles	2. Definition of Perpendicular lines
3. $\angle ASV \cong \angle ESV$	3. Right Angle Theorem
4. $S$ is a midpoint of $\overline{AE}$	4. Given
5. $\overline{ES} \cong \overline{AS}$	5. Definition of Midpoint
6. $\overline{VS} \cong \overline{VS}$	6. Reflexive Property
7. $\triangle AVS \cong \triangle EVS$	7. SAS Congruence Postulate or LL Congruence Theorem
8. $\overline{AV} \cong \overline{EV}$	8. CPCTC
9. $\triangle AVE$ is an isosceles triangle	9. Definition of Isosceles Triangle

Solution: (Complete Proof)

To justify that $\triangle AVE$ is an isosceles triangle, we are given with $\overline{VS} \perp \overline{AE}$ which is by definition the angles formed by the two segments are right angles which are $\angle ASV$ and $\angle ESV$. We are provided also with the given that $S$ is the midpoint of $\overline{AE}$ which is by definition the segment is divided into two congruent segments. Since the two triangles has common side and by reflexive property that says the segment is congruent to itself. Thus, we can use this to justify the congruence of the corresponding sides of the triangles. Now based on this we can use the SAS congruence theorem or LL Congruence to justify the congruence of the triangles. When two triangles are congruent, corresponding parts of congruent triangles are also congruent. Thus, another pair of corresponding sides are congruent that makes the triangle isosceles by definition.

We had just proven that $\triangle AVE$ is an isosceles triangle since we have established that $\overline{AV}$ and $\overline{EV}$ are two congruent sides of $\triangle AVE$ by CPCTC.

Illustrative Examples 3:

1. Given: $\angle D \cong \angle B$
   $\angle DAC \cong \angle BCA$
   Prove: $\angle ACD \cong \angle CAB$
   Proof:

[illustration: parallelogram with vertices A (top left), B (top right), C (bottom right), D (bottom left) and diagonal from A to C]

Statements	Reasons
1. _____________________ (What are the given statements?)	1. Given
2. $\overline{AC} \cong \overline{AC}$	2. Reflexive Property
3. $\triangle ADC \cong \triangle CBA$	3. _____________________ (Based on the statements, what theorem can justify $\triangle ADC \cong \triangle CBA$?)
4. $\angle ACD \cong \angle CAB$	4. _____________________ (What can justify the congruence of the corresponding parts of congruent triangles?)

$\triangle ACD$ and $\triangle CAB$ are adjacent triangles since they have a common side.

Solution: (Complete Proof)

Statements	Reasons
1. $\angle D \cong \angle B$ $\angle DAC \cong \angle BCA$	1. Given
2. $\overline{AC} \cong \overline{AC}$	2. Reflexive Property
3. $\triangle ADC \cong \triangle CBA$	3. AAS Congruence Theorem
4. $\angle ACD \cong \angle CAB$	4. CPCTC

To justify that $\angle ACD \cong \angle CAB$, we are given with $\angle D \cong \angle B$ and $\angle DAC \cong \angle BCA$. Since the two triangles have a common side, by reflexive property that says the segment is congruent to itself. Thus, we can use this to justify the congruence of the corresponding sides and the congruence of the corresponding angles of the triangles. Now based on this we can use the AAS congruence theorem to justify the congruence of the triangles. When two triangles are congruent, corresponding parts of congruent triangles are also congruent. Thus, another pair of corresponding angles are congruent by CPCTC.

Illustrative Examples 4:

Given: $\overline{DB} \cong \overline{OY}$ $\angle 1 \cong \angle 2$ Prove: $\overline{OB} \cong \overline{DY}$

Proof:

[illustration: quadrilateral with vertices B (bottom left), O (top left), D (top right), Y (bottom right); diagonals drawn crossing; angle marks labeled 1 (near B), 2 (near Y), 3 (near O), 4 (near D)]

Statements	Reasons
1. _________________ (What are the given statements?)	1. Given
2. _________________ (Which side of $\triangle BOY$ and $\triangle YDB$ is common?)	2. Reflexive Property
3. _________________ (What are the congruent triangles based on the previous statements?)	3. SAS Congruence Postulate
4. _________________ (What is to be proven?)	4. CPCTC

Solution: (Complete Proof)

Statements	Reasons
1. $\overline{DB} \cong \overline{OY}$ $\angle 1 \cong \angle 2$	1. Given
2. $\overline{BY} \cong \overline{BY}$	2. Reflexive Property
3. $\triangle BOY \cong \triangle YDB$	3. SAS Congruence Postulate
4. $\overline{OB} \cong \overline{DY}$	4. CPCTC

$\triangle BOY$ and $\triangle YDB$ are congruent by SAS Congruence Postulate and by CPCTC, $\overline{OB} \cong \overline{DY}$. We could also prove the congruency of multiple angles such as $\angle BOY \cong \angle YDB$ and $\angle OBY \cong \angle DYB$.

Illustrative Examples 5:

Given: $\overline{DL} \cong \overline{AL}$ $E$ is a midpoint at $\overline{DA}$ Prove: $\angle 3 = \angle 4$

Proof:

[illustration: triangle with apex L at top, base points D (bottom left) and A (bottom right); point E on base between D and A; segment from L to E; angle marks labeled 1 (at D), 2 (at A), 3 and 4 (at L on either side of segment LE)]

Statements	Reasons
1. _____________________ (What is the first given?)	1. Given
2. $\triangle DAL$ is an isosceles triangle	2. Definition of Isosceles Triangle
3. $\angle 1 \cong \angle 2$	3. _______________________ (Based on the previous statements, what theorem is used to deduce this statement?)

transcriber’s note: the reason cell of row 3 continues from the previous page; the table continues below.

Statements	Reasons
	what theorem is used to deduce this statement?)
4. $E$ is a midpoint at $\overline{DA}$	4. Given
5. _________________ (If $E$ is a midpoint of $\overline{DA}$, what happens to $\overline{DA}$?)	5. Definition of Midpoint
6. $\triangle DEL \cong \triangle AEL$	6. _________________ (What postulate or theorem can justify the congruency of the two triangles based on the proved statements?)
7. $\angle 3 = \angle 4$	7. _________________ (What statement can justify the other corresponding parts of congruent triangles are also congruent?)

Solution: (Complete Proof)

Statements	Reasons
1. $\overline{DL} \cong \overline{AL}$	1. Given
2. $\triangle DAL$ is an isosceles triangle	2. Definition of Isosceles Triangle
3. $\angle 1 \cong \angle 2$	3. Isosceles Triangle Theorem
4. $E$ is a midpoint at $\overline{DA}$	4. Given
5. $\overline{DE} \cong \overline{AE}$	5. Definition of Midpoint
6. $\triangle DEL \cong \triangle AEL$	6. SAS Congruence Postulate
7. $\angle 3 = \angle 4$	7. CPCTC

To justify that $\angle 3 = \angle 4$, we are given with $\overline{DL} \cong \overline{AL}$ by definition $\triangle DAL$ is an isosceles triangle and if two sides of an isosceles triangle are congruent, then the angles opposite them are also congruent. Since the two triangles has common side, by reflexive property, the segment is congruent to itself. We are also given that $E$ is a midpoint at $\overline{DA}$. By definition, the segment is divided into two congruent segments. Thus, we can use this to justify the congruence of the corresponding sides and the congruence of the corresponding angles of the triangles. Now based on this, we can use the SAS congruence theorem to justify the congruence of the triangles. When two triangles are congruent, corresponding parts of congruent triangles are also congruent. Thus, another pair of corresponding angles are congruent by CPCTC.

It is really possible to go further after proving the congruence of two triangles like proving the other corresponding parts are also congruent.

What’s More

Activity 1: Complete Me!

Given: $\overline{LO}$ bisects $\overline{KM}$ $\angle KOL \cong \angle MOL$

Prove: $\angle KLO \cong \angle MLO$

Proof:

[illustration: triangle with top side from M (left) through O (middle) to K (right); apex L at bottom below O; segment from O to L; labels M, O, K, L]

Statements	Reasons
1. ____________________ (Which of the given statement will help the next statements)	1. Given
2. $\overline{KO} \cong \overline{MO}$	2. Definition of Segment Bisector
3. ____________________ (Based on the reason given on the right, which statement will lead to the congruence of the triangles in the statement 5?)	3. Reflexive Property
4. $\angle KOL \cong \angle MOL$	4. ____________________ (Based on statement number 4, what will be the next reason?)
5. $\triangle KOL \cong \triangle MOL$	5. ____________________ (What postulate or theorem that justifies the congruency of the two triangles?)
6. $\angle KLO \cong \angle MLO$	6. ____________________ (After proving that the two triangles are congruent, what will happen to the other parts of the triangle? What is the appropriate reason for it?)

Activity 2: Fill Me!

Given: Isosceles $\triangle STA$ with respect to the vertex $\angle T$, $\overline{AR} \cong \overline{SR}$ Prove: $\angle ATR \cong \angle STR$

Proof:

[illustration: isosceles triangle with apex T at top, base corners S (bottom left) and A (bottom right); point R on base between S and A; segment from T to R; labels T, S, R, A]

Statements	Reasons
1. _____________________ (Which of the given statement will help the next statements)	1. Given
2. _____________________ (Which parts of the given triangle are congruent as defined by the isosceles triangle?)	2. Definition of Isosceles Triangle
3. _____________________ (Which side of $\triangle ATR$ and $\triangle STR$ is common?)	3. Reflexive Property
4. ________________________ (What other statement is given?)	4. Given
5. ______________________ (What are the congruent triangles based on the previous statements)	5. SSS Congruence Postulate
6. ______________________ (What other corresponding parts of the two triangles are to be proven congruent?)	6. CPCTC

Activity 3: Prove Me!

Given: $\overline{EA} \perp \overline{AC}$ $\overline{FD} \perp \overline{DB}$ $\overline{EC} \cong \overline{FB}$ $\overline{AB} \cong \overline{CD}$ Prove: $\overline{AE} \cong \overline{DF}$

[illustration: rectangle-like figure with top side from A (top left) to D (top right) and bottom side from E (bottom left) to F (bottom right); points B and C on the top side; right-angle marks at A and D; segments connecting E to B and C, and F to B and C]

Proof:

Statements	Reasons
1. $\overline{EA} \perp \overline{AC}$ and $\overline{FD} \perp \overline{DB}$	1. Given
2. _____________________ (What angle will be formed if $\overline{EA} \perp \overline{AC}$ and $\overline{FD} \perp \overline{DB}$? Name the angles)	2. Definition of Perpendicular Lines
3. $\angle EAC \cong \angle FDB$	3. _____________________ (What theorem that states that all right triangles are congruent?)
4. $\triangle EAC$ and $\triangle FDB$ are right triangles	4. Definition of Right Triangle
5. $\overline{EC} \cong \overline{FB}$ and $\overline{AB} \cong \overline{CD}$	5. Given

transcriber’s note: the table continues from the previous page.

Statements	Reasons
6. $\lvert AB \rvert = \lvert CD \rvert$	6. Definition of congruent line segments
7. $\overline{BC} \cong \overline{BC}$	7. _____________________ (What property states that a segment is always congruent to itself?)
8. $\lvert AB \rvert + \lvert BC \rvert = \lvert AC \rvert$ and $\lvert BC \rvert + \lvert CD \rvert = \lvert BD \rvert$	8. Segment Addition Postulate
9. $\lvert CD \rvert + \lvert BC \rvert = \lvert AC \rvert$ and $\lvert BC \rvert + \lvert CD \rvert = \lvert BD \rvert$	9. Substitution Property
10. $\lvert AC \rvert = \lvert BD \rvert$	10. Transitivity or Transitive Property
11. $\overline{AC} \cong \overline{BD}$	11. Definition of Congruent line segments
12. $\triangle EAC \cong \triangle FDB$	12. _____________________ (What theorem justifies that the two triangles are congruent?)
13. $\overline{AE} \cong \overline{DF}$	13. CPCTC

What I Have Learned

Directions: Fill in the blank with the correct word or phrase. Write your answer on a separate sheet of paper.

I have learned that there are two classifications of triangles; that is according to 1. _____ and according to 2. _____. Some triangles have characteristics that come from both of these classifications. Examples of them are 3. _____ and 4. _____.

I also learned that after proving two triangles are congruent, we can prove other corresponding parts of congruent triangles like proving congruence of 5. _____ and congruence of other 6. _____ of triangles by CPCTC which is also known as 7. _____.

We can also prove congruence of triangles that are 8. _____ or triangles that have common parts.

What I Can Do

Directions: Name and illustrate the objects found at home that show congruent triangles. Justify why the parts of the triangles are congruent.

Questions:

1. Were you able to name and identify objects/things that show triangle congruence?
2. Identify concepts/principles that you use in naming and illustrating the congruence of objects/things. Where do you usually apply these concepts/principles?

Rubrics:

10	8	6	4	2
Able to identify at least 5 objects showing congruent triangles and able to justify their congruence	Able to identify 4 objects showing congruent triangles and able to justify their congruence	Able to identify 3 objects showing congruent triangles and able to justify their congruence	Able to identify 2 objects showing congruent triangles and able to justify their congruence	Able to identify at least 1 object showing congruent triangles but unable to justify their congruence

Assessment

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

1. ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹 are isosceles right triangles. 𝐴𝐵̅̅̅̅ ≅ 𝐷𝐸̅̅̅̅ and 𝐴𝐶̅̅̅̅ ≅ 𝐷𝐹̅̅̅̅, which of the following statements is true by CPCTC?
   A. 𝐴𝐶̅̅̅̅ ≅ ̅𝐸𝐹̅̅̅ C. 𝐶𝐴̅̅̅̅ ≅ 𝐸𝐹̅̅̅̅
   B. 𝐵𝐶̅̅̅̅ ≅ ̅𝐸𝐹̅̅̅ D. 𝐶𝐵̅̅̅̅ ≅ 𝐹𝐷̅̅̅̅

For item 2 and 3, use the figure at the right.

2. In ∆𝐽𝑂𝑌, let S be a midpoint of 𝐽𝑌̅̅̅ and 𝑂𝑆̅̅̅̅ ⊥ 𝐽𝑌̅̅̅. What theorem or postulate can justify that ∆𝐽𝑂𝑆 ≅ ∆𝑌𝑂𝑆?
   A. AAS C. LL
   B. ASA D. HyL

[illustration: triangle JOY with point S on side JY and segment OS drawn perpendicular to JY; labels J, O, Y, S]

3. Refer to item number 2, what are the other corresponding angles that are congruent by CPCTC?
   I. ∠𝐽𝑂𝑆 ≅ ∠𝑂𝑌𝑆 II. ∠𝐽𝑂𝑆 ≅ ∠𝑌𝑂𝑆 III. ∠𝑂𝐽𝑆 ≅ ∠𝑂𝑌𝑆
   A. I only B. II only C. I and II only D. II and III only

   
   

4. In ∆𝑊𝐸𝑇 and ∆𝐷𝑅𝑌, 𝑊𝐸̅̅̅̅̅ ≅ 𝐷𝑅̅̅̅̅, 𝐸𝑇̅̅̅̅ ≅ 𝑅𝑌̅̅̅̅ and 𝑊𝑇̅̅̅̅̅ ≅ 𝐷𝑌̅̅̅̅. By SSS Congruence Postulate, we can say that ∆𝑊𝐸𝑇 ≅ ∆𝐷𝑅𝑌. Which congruence statement/s of the other corresponding parts of the two triangles is/are true?
   I. ∠𝑊 ≅ ∠𝐷 II. ∠𝐸 ≅ ∠𝑅 III. ∠𝑇 ≅ ∠𝑌
   A. I only B. II only C. II and III only D. I, II and III

   
   

5. The figure at the right is formed by overlapping triangles. Which of the following congruence statements is true?
   A. ∆𝐶𝐺𝑆 ≅ ∆𝐴𝐸𝑁
   B. ∆𝐶𝐺𝑆 ≅ ∆𝐻𝐸𝑁
   C. ∆𝑆𝐶𝐺 ≅ ∆𝐴𝐸𝑁
   D. ∆𝑆𝐶𝐺 ≅ ∆𝑁𝐸𝐻

   
   

[illustration: two overlapping triangles; vertices labeled C, A, H, S, E, G, N; tick marks on congruent sides and arcs on congruent angles]

6. Refer to item number 5, ∠𝐶𝑆𝐺 ≅ ∠𝐴𝑁𝐸 can be justified by which of the following mathematical concept?
   A. CPCTC C. Vertical Angle Theorem
   B. Right Angle Theorem D. ASA Congruence Postulate

   
   

7. Which theorem or postulate can be used to show that ∆𝐺𝐸𝐷 ≅ ∆𝐼𝐷𝐸?
   A. AAS Congruence Theorem
   B. ASA Congruence Postulate
   C. HL Congruence Theorem
   D. SAS Congruence Postulate

   
   

[illustration: rectangle GI-ED with diagonals crossing at U; labels G, I, E, D, U; angle and side markings]

8. If ∆𝐸𝐻𝑇 ≅ ∆𝐴𝑇𝐻, which congruence statement of the corresponding parts of the two triangles is true?
   A. ∠𝐻𝐸𝑇 ≅ ∠𝐴𝐻𝑇
   B. ∠𝐻𝐸𝑇 ≅ ∠𝑇𝐴𝐻
   C. ∠𝑇𝐸𝐻 ≅ ∠𝐸𝑇𝐴
   D. ∠𝑇𝐻𝐸 ≅ ∠𝑇𝐻𝐴

[illustration: figure with points H, E, T, A; right-angle marks at H and T; tick marks on congruent segments]

For items 9 to 12, use the figure at the right. Compete the proof by filling in the missing statements or reasons. Choose the letter of the correct answer from the box.

Given: 𝐴𝐵̅̅̅̅ ≅ 𝐸𝐷̅̅̅̅, 𝐴𝐶̅̅̅̅ ≅ 𝐸𝐹̅̅̅̅, 𝐵𝐹̅̅̅̅ ≅ 𝐷𝐶̅̅̅̅

[illustration: two triangles ABC and EFD sharing point G where segments AC and EF cross; labels A, B, C, D, E, F, G]

Prove: ∠𝐴𝐶𝐵 ≅ ∠𝐸𝐹𝐷

A. |𝐵𝐹| + |𝐹𝐶| = |𝐷𝐶| + |𝐶𝐹| B. 𝐺𝐹̅̅̅̅ ≅ 𝐺𝐶̅̅̅̅ C. Reflexive Property of Equality D. Transitive Property of Equality E. ∆𝐴𝐵𝐶 ≅ ∆𝐸𝐹𝐷 F. ∆𝐴𝐵𝐶 ≅ ∆𝐸𝐷𝐹

Proof:

Statements	Reasons
1. 𝐴𝐵̅̅̅̅ ≅ 𝐸𝐷̅̅̅̅ 𝐴𝐶̅̅̅̅ ≅ ̅𝐸𝐹̅̅̅ 𝐵𝐹̅̅̅̅ ≅ 𝐷𝐶̅̅̅̅	1. Given
2. 𝐹𝐶̅̅̅̅ ≅ 𝐹𝐶̅̅̅̅	2. (9)
3. 𝐹𝐶̅̅̅̅ ≅ 𝐶𝐹̅̅̅̅	3. Symmetric Property
4. (10)	4. Addition Property of Equality
5. |𝐵𝐹| + |𝐹𝐶| = |𝐵𝐶| |𝐷𝐶| + |𝐶𝐹| = |𝐷𝐹|	5. Definition of Betweenness
6. |𝐵𝐶| = |𝐷𝐹|	6. (11)
7. 𝐵𝐶̅̅̅̅ ≅ 𝐷𝐹̅̅̅̅	7. Definition of Congruent Segments
8. (12)	8. SSS Congruence Postulate
9. ∠𝐴𝐶𝐵 ≅ ∠𝐸𝐹𝐷	9. CPCTC

For items 13 to 15, use the figure at the right. Compete the proof by filling in the missing statements or reasons. Choose the letter of the correct answer form the box.

A. ̅𝐵𝐸̅̅̅ ≅ 𝐶𝐸̅̅̅̅ B. CPCTC C. Converse of Isosceles Triangle Theorem D. SAS Congruence Postulate E. Definition of Isosceles Triangle F. Isosceles Triangle Theorem

[illustration: rectangle ABCD with diagonals AC and DB meeting at E; labels A, B, C, D, E]

Given: 𝐴𝐶̅̅̅̅ ≅ 𝐷𝐵̅̅̅̅, 𝐴𝐵̅̅̅̅ ≅ 𝐷𝐶̅̅̅̅, ∠𝐵𝐴𝐶 ≅ ∠𝐶𝐷𝐵 Prove: ∆𝐵𝐸𝐶 is an isosceles triangle

Proof:

Statements	Reasons
1. 𝐴𝐶̅̅̅̅ ≅ 𝐷𝐵̅̅̅̅ 𝐴𝐵̅̅̅̅ ≅ 𝐷𝐶̅̅̅̅ ∠𝐵𝐴𝐶 ≅ ∠𝐶𝐷𝐵	1. Given
2. ∆𝐵𝐴𝐶 ≅ ∆𝐶𝐷𝐵	2. (13)
3. ∠𝐴𝐶𝐵 ≅ ∠𝐷𝐵𝐶	3. CPCTC
4. ̅𝐵𝐸̅̅̅ ≅ 𝐶𝐸̅̅̅̅	4. (14)
5. ∆𝐵𝐸𝐶 is an isosceles triangle	5. (15)

Additional Activities

Direction: Make a two-column proof based on the situation below.

[photo: folding chair viewed from the front, with diagonal legs crossing at point H; corners labeled C, A, R, I]

The front diagonal legs of the fordable chair are congruent and are joined at their midpoints, 𝐶𝐼̅̅̅ ≅ 𝐴𝑅̅̅̅̅. If we draw a line from the corner of the chair to both of the front foot caps, we will have 𝐶𝑅̅̅̅̅ and 𝐴𝐼̅̅̅ perpendicular to 𝑅𝐼̅̅̅. Prove that ∠𝑅𝐶𝐼 ≅ ∠𝐼𝐴𝑅.

Proof:

Statements	Reasons
1.	1.
2.	2.
3.	3.
4.	4.
5.	5.
6.	6.
7	7
8.	8.

Question:

1. What are other corresponding parts of ∆𝐶𝑅𝐼 and ∆𝐴𝐼𝑅 that can be proven congruent also by CPCTC?

Rubrics:

10	8	6	4	2
Able to provide correct, logical, 2 column proof	Able to provide statements and reasons but with 1 erroneous statements or reasons	Able to provide statements and reasons but with 2 erroneous statements or reasons	Able to provide statements and reasons but with at least 3 erroneous statements or reasons	Able to provide only the given statements

Answer Key

transcriber’s note: this page is printed rotated 180 degrees. The text has been read in upright order below. Several cells in the What’s New, What’s More, and Additional Activities tables are hard to read at this resolution; those cells are marked accordingly.

What I Know

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

What’s In

1. H
2. E
3. F
4. C
5. G
6. D
7. A
8. B

What’s New

I.

Statements	Reasons
1. Square CARE and its diagonal C̅R̅	1. Given [?]
2. [illegible]	2. [illegible]
3. C̅R̅ ≅ C̅R̅	3. [illegible]
4. [illegible]	4. SSS Congruence Postulate

II.

Statements	Reasons
1. [illegible]	1. Given
2. [illegible]	2. Given
3. [illegible]	3. Definition of Midpoint
4. [illegible]	4. Definition of Vertical angles
5. [illegible]	5. Vertical Angle Theorem
6. [illegible]	6. ASA Congruence Postulate

What’s More: Act 1

Statements	Reasons
1. L̅O̅ bisects K̅M̅	1. [illegible]
2. [illegible]	2. [illegible]
3. L̅O̅ ≅ L̅O̅	3. Given [?]
4. [illegible]	4. [illegible]
5. [illegible]	5. SAS Congruence Postulate
6. [illegible]	6. CPCTC

What’s More: Act 2

Statements	Reasons
1. ∆STA with vertex ∠T is isosceles	1. Isosceles [?]
2. S̅T̅ ≅ A̅T̅	2. [illegible]
3. T̅R̅ ≅ T̅R̅	3. [illegible]
4. A̅R̅ ≅ S̅R̅	4. [illegible]
5. ∆ATR ≅ ∆STR	5. [illegible]
6. ∠ATR ≅ ∠STR	6. CPCTC

What I Can Do

(Students answer varies.)

Assessment

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

Additional Activities

Statements	Reasons
1. The front diagonal legs of the fordable chair are congruent and are joined at their midpoints, C̅I̅ ≅ A̅R̅	1. Given
2. A̅I̅ and C̅R̅ perpendicular to R̅I̅	2. Given
3. ∠AIR and ∠CRI are right angles	3. Definition of Perpendicular Lines
4. ∠AIR ≅ ∠CRI	4. Right Angle Theorem
5. ∆AIR and ∆CRI are right triangles	5. Definition of Right Triangle
6. R̅I̅ ≅ R̅I̅	6. Reflexive Property
7. ∆AIR ≅ ∆CRI	7. HyL Congruence Theorem
8. ∠IAR ≅ ∠RCI	8. CPCTC

What’s More: Act 3

Statements	Reasons
1.	1.
2. ∠FDA and ∠EAD are right angles	2. Given [?]
3.	3. Right Angle Theorem
4.	4.
5.	5.
6.	6. Reflexive Property
7.	7.
8.	8.
9.	9.
10.	10. HyL Congruence
11.	11.

What I Have Learned

I have learned that there are two classifications of triangles, that is according to sides. There are some triangles that contain characteristics that come from both of these classifications. Few of them are isosceles right triangle and scalene right triangle.

I also learned that in proving two triangles are congruent, we can also prove the other corresponding parts as congruent by CPCTC. CPCTC is also known as Corresponding Parts of Congruent Triangles are Congruent. We can prove congruent angles and other sides of multiple triangles, like proving triangles that are overlapped. Overlapping triangles are triangles that have common sides or angles.

References

A. Book

1. Emmanuel Abunzo, et. al., Mathematics 8 Learner’s Module (Pasig City, Philippines: Book Media Press, Inc. and Prinwell, Inc., 2013).
2. A Baccay, Geometry for Secondary Schools (Philippines: Phoenix Publishing House), 140-190.
3. Ricardo Crisostomo, Priscila de Sagun, and Alicia Padua, Our World of Math (Quezon City, Philippines: Vibal Publishing House, Inc., 2013).
4. Soledad Dilao and Julieta Bernabe, Geometry (Quezon City, Philippines: SD Publication, Inc., 2010).
5. Robert Kalin and Mary Kay Corbitt, Geometry, Prentice Hall (Jurong, Singapore: Pearson Education, Inc., 1993).
6. Karl Romero and Freidrich Jose, Geometry in the Real World: Explorations and Applications (Makati City, Philippines: Salesiana Publishers, Inc., 2003).

B. Electronic Resources

1. www.mathbitnotebook.com/Geometry/BasicTerms/BTauxiliary.html

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