Mathematics 8 Quarter 4 Module 3: Proving Inequalities in a Triangle

Mathematics 8 Quarter 4 – Module 3: Proving Inequalities in a Triangle

Mathematics Quarter 4 – Module 3 Proving Inequalities in a Triangle

[illustration: students and a teacher at a table with a globe, a child reading by a lantern, and students reading under a tree]

CO_Q4_Mathematics 8_ Module 3

Mathematics – Grade 8 Alternative Delivery Mode Quarter 4 – Module 3 Proving Inequalities in a Triangle First Edition, 2020

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Development Team of the Module

Writers: Melody C. Gapa and Ella C. Armayan

Language Editor: Wilmar N. Espinosa

Content Evaluators: Victorino Nimes, Myracell Buenaflor, and Marie Grace O. Aparre

Layout Evaluator: Devina P. Malinao

Reviewers: Rhea J. Yparraguirre, Lewellyn V. Mejias, Mercedita D. Gonzaga and Juliet P. Utlang

Illustrators: Melody C. Gapa and Ella C. Armayan

Layout Artists: Melody C. Gapa, Ella C. Armayan and 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

8

Mathematics Quarter 4 – Module 3 Proving Inequalities in a Triangle

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 master the skills of proving inequalities in a triangle. 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 on proving inequalities in a triangle (M8GE-IVc-1).

After going through this module, you are expected to:

1. state the theorems on exterior angle inequality, triangle inequality, Hinge and its converse,
2. prove statements involving triangle inequalities using theorems on inequalities in triangles, and
3. justify the importance of proving triangle inequalities in real life situation.

What I Know

Directions: Read and answer each of the questions carefully and choose the letter of the correct answer. Write your answer in a separate sheet of paper.

[illustration: triangle with vertices C (top), E (bottom left), A (bottom right); side CE labeled 4 cm, side CA labeled 3 cm, side EA labeled 5 cm; angle at A marked with an arc]

1. Which of the following is the correct ascending arrangement of the measures of the angles of ∆CEA?
   A. ∠𝐸, ∠𝐴, ∠𝐶
   B. ∠𝐴, ∠𝐶, ∠𝐸
   C. ∠𝐸, ∠𝐶, ∠𝐴
   D. cannot be determined

   
   

2. Which property of equality is defined by the statement “For all real numbers 𝑝, 𝑝 = 𝑝”?
   A. Reflexive Property
   B. Transitive property
   C. Symmetric Property
   D. Substitution Property

   
   

3. Which statement below defines Addition Property of Equality?
   A. For all real numbers, 𝑝, 𝑞 and 𝑟, if 𝑝 = 𝑞, then 𝑝 + 𝑟 = 𝑞 + 𝑟.
   B. For all real numbers, 𝑝, 𝑞 and 𝑟, if 𝑞 = 𝑟, then 𝑝 + 𝑟 = 𝑞 + 𝑟.
   C. For all real numbers, 𝑝, 𝑞 and 𝑟, if 𝑝 = 𝑞, then 𝑝 + 𝑞 = 𝑞 + 𝑟.
   D. For all real numbers, 𝑝, 𝑞 and 𝑟, if 𝑝 = 𝑟, then 𝑝 + 𝑟 = 𝑞 + 𝑟.

   
   

4. Which of the following theorems states that “base angles of isosceles triangles are congruent”?
   A. Linear Pair Theorem
   B. Exterior Angle Theorem
   C. Vertical Angles Theorem
   D. Isosceles Triangle Theorem

   
   

5. What triangle inequality theorem states that “If one side of a triangle is longer than a second side, then the angle opposite the longer side is larger than the angle opposite the second side”?
   A. Exterior Angle Inequality Theorem
   B. Triangle Inequality Theorem 1 (Ss→Aa)
   C. Triangle Inequality Theorem 2 (Aa→Ss)
   D. Triangle Inequality Theorem 3 (S1 + S2 > S3)

   
   

6. Which statement defines Exterior Angle Inequality Theorem?
   A. The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
   B. The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
   C. If one side of a triangle is longer than a second side, then the angle opposite the longer side is larger than the angle opposite the second side.

   
   

D. If one angle of a triangle is larger than a second angle, then the side opposite the larger angle is longer than the side opposite the second angle.

7. Which theorem below does not belong to the inequality theorems in one triangle?
   A. Hinge Theorem
   B. Exterior Angle Inequality Theorem
   C. Triangle Inequality Theorem 1 (Ss→Aa)
   D. Triangle Inequality Theorem 2 (Aa→Ss)

   
   

8. Which of the following statements is NOT true about writing proofs?
   A. It does not make use of hints to aid one’s thinking.
   B. Guide questions are provided to help one succeed in the next activities.
   C. It will develop observation skills, deductive thinking and logical reasoning.
   D. There is a need to determine the appropriate statements and give reasons behind these statements.

   
   

9. Given ∆NKE and ∆ICA below, if 𝑚∠𝐾 > 𝑚∠𝐶, using the Hinge Theorem, which could be the conclusion about the given figures?
   A. |𝐸𝑁| > |𝐴𝐼|
   B. |𝐸𝑁| < |𝐴𝐼|
   C. 𝑚∠𝐸 < 𝑚∠𝐴
   D. 𝑚∠𝑁 < 𝑚∠𝐼

   
   

[illustration: triangle NKE with vertices K (top left), N (top right), E (bottom); triangle ICA with vertices C (top left), I (top right), A (bottom); tick marks show congruent sides]

10. With the use of the Converse of the Hinge Theorem, which of the statements below is true about the sides and angles of ∆𝑀𝐴𝑁 and ∆𝐵𝑂𝑌?
    A. If 𝑀𝐴 ≅ 𝐵𝑂, 𝑀𝑁 ≅ 𝐵𝑌 and |𝐴𝑁| >|𝑂𝑌|, then 𝑚∠𝑀 > 𝑚∠𝐵
    B. If 𝑀𝐴 ≅ 𝐵𝑂, 𝑀𝑁 ≅ 𝐵𝑌 and |𝐴𝑁| > |𝑂𝑌|, then 𝑚∠𝑀 < 𝑚∠𝐵
    C. If 𝑀𝐴 ≅ 𝐵𝑂, 𝑀𝑁 ≅ 𝐵𝑌 and |𝐴𝑁| > |𝑂𝑌|, then 𝑚∠𝐴 > 𝑚∠𝑂
    D. If 𝑀𝐴 ≅ 𝐵𝑂, 𝑀𝑁 ≅ 𝐵𝑌 and |𝐴𝑁| > |𝑂𝑌|, then 𝑚∠𝑁 > 𝑚∠𝑌

[illustration: triangle MAN with vertices A (top), M (bottom left), N (bottom right) and triangle BOY with vertices O (top), B (bottom left), Y (bottom right); tick marks show congruent sides]

11. Given ∆𝑆𝐴𝑁 at the right, which one supports the statement 𝑚∠𝑆𝐴𝑁 = 𝑚∠1 + 𝑚∠3?
    A. Linear Pair Postulate
    B. Substitution Property
    C. Property of Inequality
    D. Angle Addition Postulate

[illustration: triangle SAN with vertices S (top), A (bottom left), N (bottom right); point D on side SN; segment AD drawn; angles numbered 1 (at A, between AS and AD), 2 (at D), and 3 (at A, between AD and AN); tick marks show congruent sides]

12. Ryan is analyzing the pair of triangles in the figure below. Which of the following is the correct observation?
    A. 𝑅𝐴 ≅ 𝐶𝐸
    B. |𝐶𝐸| > |𝑅𝐸|
    C. 𝑚∠𝑅𝐸𝐴 < 𝑚∠𝐶𝐸𝐴
    D. 𝑚∠𝑅𝐸𝐴 > 𝑚∠𝐶𝐸𝐴

[illustration: quadrilateral with vertices A (top), C (left), R (right), E (bottom), with segment AE drawn; side AC labeled 10, side AR labeled 11, side CE labeled 6, side ER labeled 6]

13. Which of the following would justify Ryan’s correct observation in item no. 12?
    A. Hinge Theorem
    B. Converse of Hinge Theorem
    C. Triangle Inequality Theorem 1 (Ss→Aa)
    D. Triangle Inequality Theorem 3 (S1 + S2 > S3)

    
    

14. From the triangles shown below, a conclusion can be reached using the converse of hinge theorem. Which of the following is the conclusion?
    A. 𝑆𝐼 ≅ 𝑆𝑇
    B. |𝑆𝐼| < |𝑆𝐴|
    C. 𝑚∠𝐼𝑆𝑇 > 𝑚∠𝐴𝑆𝑇
    D. 𝑚∠𝐴𝑆𝑇 > 𝑚∠𝐼𝑆𝑇

    
    

[illustration: quadrilateral with vertices I (top), S (left), T (right), A (bottom), with segment ST drawn; side SI labeled 12, side IT labeled 9, side TA labeled 7, side SA labeled 12]

15. In going to school, Ianne concluded that it will take him longer to walk through Irizari and Tuldanes Streets. He chooses to walk instead through Herrera St. daily in order to arrive early. Which triangle inequality theorem justifies his choice?
    A. Exterior Angle Inequality Theorem
    B. Triangle Inequality Theorem 1 (Ss→Aa)
    C. Triangle Inequality Theorem 2 (Aa→Ss)
    D. Triangle Inequality Theorem 3 (S1 + S2 > S3)

[chart: route map; routes from School to Ianne’s Residence labeled Tuldanes St. (upper route), Irizari St. (upper right route), and Herrera St. (lower route); School and Ianne’s Residence are marked with building icons]

Lesson 1: Proving Inequalities in a Triangle

In your previous lessons, you have investigated and discovered different inequality theorems in a triangle or pair of triangles which consist of the side-angle, angle-side inequality theorems, exterior angle inequality theorem, and the Hinge theorem and its converse. These inequality theorems were discussed, illustrated and their applications were explained.

As we go through this module, think about how those theorems can be proven and how important writing proofs in learning geometry is. The skills you will gain in writing proofs will enable you to justify inequalities in triangles and in triangular designs and structures that are found in the things around us.

What’s In

[illustration: hand holding a puzzle piece]

Activating Prior Knowledge!

I. Directions: Identify the postulate or property defined in each of the following statements. Choose your answer from the given options A to F. Write only the letter of the correct answer in a separate sheet of paper.

A. Angle Addition Postulate B. Segment Addition Postulate C. Addition Property of Equality D. Reflexive Property of Equality E. Transitive Property of Equality F. Substitution Property of Equality

1. For all real numbers 𝑝, 𝑞 and 𝑟, if 𝑝 = 𝑞 and 𝑞 = 𝑟, then 𝑝 = 𝑟.
2. If point 𝑆 lies in the interior of ∠𝑃𝑄𝑅, then 𝑚∠𝑃𝑄𝑆 + 𝑚∠𝑆𝑄𝑅 = 𝑚∠𝑃𝑄𝑅
3. If points 𝑃, 𝑄 ad 𝑅 are collinear (𝑃 − 𝑄 − 𝑅) and 𝑄 is between points 𝑃 and 𝑅, then |𝑃𝑄| + |𝑄𝑅| = |𝑃𝑅|
4. For all real numbers 𝑝 and 𝑞, if 𝑝 = 𝑞, then 𝑞 can be used to substitute 𝑝 in any expression.
5. For all real numbers p, q and r if 𝑝 = 𝑞, then 𝑝 + 𝑟 = 𝑞 + 𝑟.

II. Directions: Match each figure in Column A to the theorem it illustrates in column B. Write the letter that corresponds to the correct answer in a separate sheet of paper.

Column A	Column B
_____1. [illustration: triangle STI with two congruent sides marked at I; labeled S, I, T] 𝑚∠𝑇 > 𝑚∠𝑆	A. Hinge Theorem
_____2. [illustration: triangle AVN with vertices A, V, N] |𝑉𝐴| + |𝐴𝑁| > |𝑉𝑁|	B. Converse of Hinge Theorem
_____3. [illustration: line RK with point O on it and point C above; labeled R, O, K, C] 𝑚∠𝐶𝑂𝑅 > 𝑚∠𝑂𝐶𝐾	C. Triangle Inequality Theorem 1 (Ss→Aa)
_____4. [illustration: triangles INK and CAT with tick marks; labeled I, N, K, A, C, T] Given: 𝑚∠𝐶 > 𝑚∠𝑁 |𝐴𝑇| > |𝐼𝐾|	D. Triangle Inequality Theorem 2 (Aa→Ss)
_____5. [illustration: triangles DFE and GIH with tick marks; labeled D, F, E, G, I, H] Given: |𝐷𝐸| > |𝐺𝐻| 𝑚∠𝐹 > 𝑚∠𝐼	E. Triangle Inequality Theorem 3 (S1 + S2 > S3)
	F. Exterior Angle Inequality Theorem

What’s New

[illustration: hand holding a sprouting plant]

Discovering!

This part consists of proving activities for the 6 theorems on triangle inequalities. Analyze and use the hint provided in each activity to complete the proof. Note that inequalities in triangles, even without actual measurements, can be justified deductively using theorems on inequalities in triangles.

Activity 1

Directions: Complete the proof of Triangle Inequality Theorem 1(Ss→Aa) in two-column form by choosing the right statement written in the box below. Write your answer in a separate sheet of paper.

• ∠1 ≅ ∠2
• 𝑃𝑄̅̅̅̅̅ ≅ 𝑃𝑆̅̅̅̅̅
• Angle Addition Postulate
• 𝑚∠2 > 𝑚∠𝑄𝑅𝑃
• 𝑚∠𝑃𝑄𝑅 > 𝑚∠𝑃𝑅𝑄
• Property of Inequality
• 𝑚∠𝑄𝑆𝑅 + 𝑚∠𝑃𝑅𝑄 + 𝑚∠3 = 180⁰
• 𝑚∠2 = 𝑚∠𝑃𝑅𝑄 + 𝑚∠3
• ∆𝑃𝑄𝑆 is isosceles triangle
• Substitution Property

Given: ∆𝑃𝑄𝑅; |𝑃𝑅| > |𝑃𝑄| Prove: 𝑚∠𝑄 > 𝑚∠𝑅

[illustration: triangle PQR with tick marks on sides PQ and PR; labeled P, Q, R] [illustration: triangle PQR with point S on side PR, segment QS drawn, angles numbered 1, 2, 3; labeled P, Q, R, S]

Proof:

We cannot directly prove that 𝑚∠𝑄 > 𝑚∠𝑅, thus, there is a need to make additional constructions (see the second figure). Locate S on 𝑃𝑅̅̅̅̅ such that |𝑃𝑆| = |𝑃𝑄|, and connect 𝑆 𝑡𝑜 𝑄 with a segment to form a triangle PQS.

Statements	Reasons
1. _____ (How do you describe the relationship between 𝑃𝑄̅̅̅̅̅ and 𝑃𝑆̅̅̅̅? )	1. By construction
2. _____ (Based on statement 1, what kind of a triangle is ∆𝑃𝑄𝑆?)	2. Definition of isosceles triangle
3. _____ (Based on statement 1, how do you describe the relationship between ∠1 and ∠2?)	3. Base angles of isosceles triangles are congruent.
4. 𝑚∠𝑄 = 𝑚∠1 + 𝑚∠3	4. _____ (What postulate supports the statement that the sum of the measures of ∠1 and ∠3 is equal to measure of ∠𝑃𝑄𝑅?)
5. 𝑚∠𝑄 > 𝑚∠1	5. _____

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

Statements	Reasons
	(What property supports the inequality statement focusing on ∠1 based on statement 4?)
6. 𝑚∠𝑄 > 𝑚∠2	6. _____ (What property supports the step where the right side of the inequality in statement 5 is replaced with its equivalent in statement 3?)
7. _____ (Based on the illustration, write an operation statement involving measures of ∠𝑄𝑆𝑅, ∠𝑅, 𝑎𝑛𝑑∠3.	7. The sum of the measures of the interior angles of a triangle is 180⁰
8. 𝑚∠2+ 𝑚∠𝑄𝑆𝑅 = 180⁰	8. Linear pair theorem
9. 𝑚∠2+ 𝑚∠𝑄𝑆𝑅 = 𝑚∠𝑄𝑆𝑅+ 𝑚∠𝑅 + 𝑚∠3	9. Substitution/ Transitive Property
10. _____ (What will be the result if 𝑚∠𝑄𝑆𝑅 is subtracted from both sides of statement 9?)	10. Subtraction Property
11. _____ (Based on statement number 10, write an inequality statement focusing on ∠𝑅.)	11. Property of Inequality
12. _____ (Based on statements 6 and 11: If 𝑚∠𝑃𝑄𝑅 > 𝑚∠2 and 𝑚∠2 > 𝑚∠𝑅, then __)	12. Transitive Property

Activity 2

Directions: Complete the proof of Triangle Inequality Theorem 2 (Aa→Ss) in two-column form by choosing the right statement or words written in the box below. Write your answer on a separate sheet of paper.

• ∠𝑃 ≅ ∠𝑅
• Equal
• True
• 𝑚∠𝑃 > 𝑚∠𝑅
• Larger
• False
• 𝑚∠𝑃 > 𝑚∠𝑄
• ∠𝑃 ≅ ∠𝑄
• Smaller
• 𝑚∠𝑃 < 𝑚∠𝑅
• Triangle Inequality Theorem 1
• Hinge Theorem

[illustration: triangle PQR with vertices P (top), Q (bottom left), R (bottom right)]

Given: ∆𝑃𝑄𝑅; 𝑚∠𝑃 > 𝑚∠𝑅 Prove: |𝑄𝑅| > |𝑃𝑄|

To prove that |𝑄𝑅| > |𝑃𝑄|, we will use indirect proof. Thus, we need to assume that |𝑄𝑅| ≯ |𝑃𝑄|.

Proof:

Statements	Reasons
1. |𝑄𝑅| = |𝑃𝑄| 𝑜𝑟 |𝑄𝑅| < |𝑃𝑄|	1. Assumption that |𝑄𝑅| ≯ |𝑃𝑄|
2. Consider |𝑄𝑅| = |𝑃𝑄|. If |𝑄𝑅| = |𝑃𝑄|, then ∆𝑃𝑄𝑅 is an isosceles triangle. (If |𝑄𝑅| = |𝑃𝑄|, then what kind of triangle is ∆𝑃𝑄𝑅?)	2. Definition of isosceles triangle
3. _____ (Consequently, what can you say about ∠𝑃 and ∠𝑅?)	3. Base angles of isosceles triangles are congruent.
4. The assumption that |𝑄𝑅| = |𝑃𝑄| is false.	4. The conclusion that ∠𝑃 ≅ ∠𝑅 contradicts the given that 𝑚∠𝑃 > 𝑚∠𝑅
5. Consider |𝑄𝑅| < |𝑃𝑄|. If |𝑄𝑅| < |𝑃𝑄|, then _____.	5. _____ (By what theorem of triangle inequality?)
6. _____ (The assumption that 𝑄𝑅 < 𝑃𝑄 is true or false?)	6. The conclusion that 𝑚∠𝑃 < 𝑚∠𝑅 contradicts the given that 𝑚∠𝑃 > 𝑚∠𝑅
7. _____ (Therefore, 𝑄𝑅 > 𝑃𝑄 must be true or false?)	7. The assumption that |𝑄𝑅| ≯ |𝑃𝑄| contradicts the known fact that 𝑚∠𝑃 > 𝑚∠𝑅

Activity 3

Direction: Complete the proof of Triangle Inequality Theorem 3 (S1 + S2 > S3) in two-column form by choosing the right statement written in the box below. Use the previous theorem to prove the next theorem. Write your answer on a separate sheet of paper. Enjoy!

• ∠𝑃𝑅𝑆 ≅ ∠𝑃𝑆𝑅
• 𝑃𝑆̅̅̅̅ ≅ 𝑃𝑅̅̅̅̅̅
• Transitive Property
• 𝑃𝑄 + 𝑃𝑆 > 𝑄𝑅
• ∠𝑃𝑅𝑆 ≅ ∠𝑄𝑆𝑅
• Angle Addition Postulate
• Substitution Property
• |𝑃𝑄| + |𝑃𝑅| > |𝑄𝑅|
• ∆𝑃𝑅𝑆 is equilateral triangle
• ∆𝑃𝑅𝑆 is isosceles triangle
• ∆𝑃𝑅𝑆 is scalene triangle
• 𝑚∠𝑆𝑅𝑄 = 𝑚∠𝑄𝑅𝑃 + 𝑚∠𝑆𝑅𝑃
• Triangle Inequality Theorem 1 (Ss→ Aa)
• Reflexive Property of Equality
• Triangle Inequality Theorem 2 (Aa→Ss)

Given: ∆𝑃𝑄𝑅 where |𝑃𝑄| < |𝑃𝑅| < |𝑄𝑅| Prove: |𝑃𝑄| + |𝑃𝑅| > |𝑄𝑅| |𝑃𝑄| + |𝑄𝑅| > |𝑃𝑅| |𝑄𝑅| + |𝑃𝑅| > |𝑃𝑄|

[illustration: triangle PQR with vertices P (top), Q (bottom left), R (bottom right)] [illustration: triangle PQS with point P on side QS, tick marks, angles numbered 1, 2, 3; labeled P, Q, R, S]

Proof:

Since |𝑄𝑅| > |𝑃𝑅| and |𝑄𝑅| > |𝑃𝑄|, then |𝑄𝑅| + |𝑃𝑄| > |𝑃𝑅| and |𝑄𝑅| + |𝑃𝑅| > |𝑃𝑄| are true. Hence, what remains to be proven is the last statement, |𝑃𝑄| + |𝑃𝑅| > |𝑄𝑅|.

To prove |𝑃𝑄| + |𝑃𝑅| > |𝑄𝑅|, let us construct 𝑃𝑆̅̅̅̅ as an extension of 𝑃𝑄̅̅̅̅ such that 𝑃 is between 𝑄 and 𝑆, 𝑃𝑆̅̅̅̅ ≅ 𝑃𝑅̅̅̅̅̅ and ∆𝑃𝑅𝑆 is formed.

Statements	Reasons
1. _____ (What is the relationship between 𝑃𝑆̅̅̅̅ and 𝑃𝑅̅̅̅̅̅?)	1. By construction
2. _____ (Based on statement 1, what kind of a triangle is ∆𝑃𝑅𝑆?)	2. Definition of isosceles triangle
3. _____ (Based on statement 1, what is the relationship between ∠𝑃𝑅𝑆 and ∠𝑃𝑆𝑅?)	3. Base angles of isosceles triangles are congruent.
4. Based on the illustration, ∠𝑃𝑆𝑅 ≅ ∠ 𝑄𝑆𝑅	4. _____ (By what property of equality?)
If ∠𝑃𝑅𝑆 ≅ ∠𝑃𝑆𝑅 (statement 3) and ∠𝑃𝑆𝑅 ≅ ∠𝑄𝑆𝑅(statement 4), then 5. _____	5. Transitive Property of Equality
6. From the illustration, 𝑚∠𝑄𝑅𝑆 = 𝑚∠𝑃𝑅𝑄 + 𝑚∠𝑃𝑅𝑆	6. _____ (By what postulate?)
7. Based on statements 5 and 6, 𝑚∠𝑄𝑅𝑆 = 𝑚∠𝑃𝑅𝑄 + 𝑚∠𝑄𝑆𝑅	7. _____ (By what property?)
8. Based on statement 7, 𝑚∠𝑄𝑅𝑆 > 𝑚∠𝑄𝑆𝑅	8. Property of Inequality
9. |𝑄𝑆| > |𝑄𝑅|	9. _____ (By what theorem of triangle inequality?)
10. |𝑃𝑄| + |𝑃𝑆| = |𝑄𝑆|	10. Segment Addition Postulate
11. _____ (Write a statement using statements 9 and 10.)	11. Substitution Property
12. _____ (Write a statement using statements 1 and 11.)	12. Substitution Property

Activity 4

Directions: Complete the proof of Exterior Angle Inequality Theorem 1 in two-column form by choosing the right statement written in the box below. Write your answer in a separate sheet of paper.

• 𝑚∠1 + 𝑚∠2
• ∠1 ≅ ∠2
• ∠3 ≅ ∠4
• ∠𝑃𝑅𝑆 ≅ ∠𝑄𝑃𝑅
• ∠𝑄𝑃𝑅 ≅ ∠1
• ∆𝑃𝑇𝑄 ≅ ∆𝑅𝑇𝑈
• 𝑚∠𝑃𝑅𝑆 > 𝑚∠𝑄𝑃𝑅

Given: ∆𝑃𝑄𝑅 with exterior angle ∠𝑃𝑅𝑆 Prove: 𝑚∠𝑃𝑅𝑆 > 𝑚∠𝑄𝑃𝑅

[illustration: triangle PQR with side RS extended beyond R; labeled P, Q, R, S] [illustration: triangle PQR with side RS extended; point T on PR and segment QU drawn, angles numbered 1, 2, 3, 4; tick marks; labeled P, Q, R, S, T, U]

Proof:

Let us prove that 𝑚∠𝑃𝑅𝑆 > 𝑚∠𝑄𝑃𝑅 by constructing the following:

• midpoint 𝑇 on 𝑃𝑅̅̅̅̅̅ such that 𝑃𝑇̅̅̅̅̅ ≅ 𝑅𝑇̅̅̅̅̅
• 𝑄𝑈̅̅̅̅̅ through T such that 𝑄𝑇̅̅̅̅̅ ≅ 𝑇𝑈̅̅̅̅̅

Statements	Reasons
1. 𝑃𝑇̅̅̅̅̅ ≅ 𝑅𝑇̅̅̅̅̅ ; 𝑄𝑇̅̅̅̅ ≅ 𝑇𝑈̅̅̅̅	1. By construction
2. _____ (What is the relationship between ∠3 and ∠4?)	2. Vertical Angles are congruent
3. _____ (Based on statements 1 and 2, relate two triangles in the illustration)	3. SAS Triangle Congruence Postulate
4. ∠𝑄𝑃𝑅 ≅ _______. (Based on question number 2)	4. Corresponding parts of congruent triangles are congruent
5. 𝑚∠𝑃𝑅𝑆 = _______. (Based on the illustration)	5. Angle Addition Postulate
6. 𝑚∠𝑃𝑅𝑆 > 𝑚∠1	6. Property of Inequality
7. _____ (Using statement in question 3 and 𝑚∠𝑃𝑅𝑆 > 𝑚∠1)	7. Substitution Property

Activity 5

Directions: Complete the proof of Hinge Theorem or SAS triangle inequality theorem 1 in two-column form by choosing the right statement written in the box below. Write your answer/result on a separate sheet of paper. Enjoy!

• |𝐻𝑄| + |𝑅𝑄| > |𝐻𝑅|
• Property of Inequality
• |𝐻𝑄| + |𝑅𝑄| = |𝐻𝑅|
• Segment Addition Postulate
• |𝐻𝑄| + |𝑅𝑄| < |𝐻𝑅|
• Triangle Inequality Theorem 1
• Triangle Inequality Theorem 2
• Definition of Congruent Triangles
• Substitution Property of Equality

Given: ∆𝐻𝑂𝑇 and ∆𝑃𝐼𝐸; 𝐻𝑂̅̅̅̅̅ ≅ 𝑃𝐼̅̅̅̅, 𝑂𝑇̅̅̅̅ ≅ 𝐼𝐸̅̅̅̅, 𝑚∠𝐻𝑂𝑇 > 𝑚∠𝑃𝐼𝐸 Prove: |𝐻𝑇| > |𝑃𝐸|

[illustration: triangle HOT with tick marks and angle at O; triangle PIE with tick marks and angle at I; labeled H, O, T, P, I, E] [illustration: combined figure of triangles HOT and PIE with auxiliary points R, Q, S; labeled H, R, O, S, Q, T, P, E, I]

Proof:

1. Locate 𝑆 on the same side of 𝑂𝑇⃡ as 𝐻 such that ∠𝐻𝑂𝑆 ≅ ∠𝑃𝐼𝐸.
   • Consider point 𝑅 on 𝑂𝑆⃡ so that 𝑂𝑅̅̅̅̅̅ ≅ 𝑂𝑇̅̅̅̅ ≅ 𝐼𝐸̅̅̅
   • Consequently, ∆𝐻𝑂𝑅 ≅ ∆𝑃𝐼𝐸 by SAS Triangle Congruence Postulate.
2. Construct the bisector 𝑂𝑄̅̅̅̅ of ∠𝑇𝑂𝑅 such that 𝑄 is on 𝐻𝑇̅̅̅̅̅
   • ∠𝑇𝑂𝑄 ≅ ∠𝑅𝑂𝑄
   • Consequently, ∆𝑇𝑂𝑄 ≅ ∆𝑅𝑂𝑄 by SAS Triangle Congruence Postulate because 𝑂𝑄̅̅̅̅ ≅ 𝑂𝑄̅̅̅̅ by reflexive property of equality and 𝑂𝑅̅̅̅̅̅ ≅ 𝑂𝑇̅̅̅̅̅ from construction no.1. So, 𝑅𝑄̅̅̅̅̅ ≅ 𝑄𝑇̅̅̅̅̅ because corresponding parts of congruent triangles are congruent (CPCTC).

Statements	Reasons
1. |𝐻𝑇| = |𝐻𝑄| + |𝑄𝑇| (Based on the illustration)	1. _____ (By what postulate?)
2. |𝐻𝑇| = |𝐻𝑄| + |𝑅𝑄|	2. _____ (By what property of equality?)
3. In ∆𝐻𝑄𝑅, _____	3. Triangle Inequality Theorem 3
4. |𝐻𝑇| > |𝐻𝑅| (Using statements 2 and 3)	4. _____ (By what property of inequality)
5. |𝐻𝑅| = |𝑃𝐸|	5. _____ (Consider ∆𝐻𝑂𝑅 𝑎𝑛𝑑 ∆𝑃𝐼𝐸)
6. |𝐻𝑇| > |𝑃𝐸|	6. _____ (_____________________)

Activity 6

Directions: Complete the proof of the Converse of Hinge Theorem or SSS Triangle Inequality Theorem in two-column form by choosing the right statement written in the box below. Write your answer/result on a separate sheet of paper. Enjoy!

• 𝑚∠𝐹𝑈𝑁 ≯ 𝑚∠𝐴𝑅𝑇
• 𝑚∠𝐹𝑈𝑁 > 𝑚∠𝐴𝑅𝑇
• 𝑚∠𝐹𝑈𝑁 = 𝑚∠𝐴𝑅𝑇
• AAS
• SAS
• ASA
• Hinge Theorem
• True
• False

Given: ∆𝐹𝑈𝑁 and ∆𝐴𝑅𝑇 𝐹𝑈̅̅̅̅̅ ≅ 𝐴𝑅̅̅̅̅̅, 𝑈𝑁̅̅̅̅ ≅ 𝑅𝑇̅̅̅̅̅, |𝐹𝑁| > |𝐴𝑇| Prove: 𝑚∠𝑈 > 𝑚∠𝑅

[illustration: triangle FUN with tick marks on sides FU and UN; triangle ART with tick marks on sides AR and RT; labeled F, U, N, R, A, T]

Proof:

Let us use indirect proof to show that 𝑚∠𝑈 > 𝑚∠𝑅. Thus, we will assume that 𝑚∠𝑈 ≯ 𝑚∠𝑅.

Hinge Theorem or SAS Inequality Theorem

If two sides of one triangle are congruent to two sides of another triangle, but the included angle in the first triangle is greater than the included angle in the second, then the third side of the first triangle is longer than the third side of the second.

Given: ∆𝐻𝑂𝑇 and ∆𝑃𝐼𝐸; 𝐻𝑂̅̅̅̅ ≅ 𝑃𝐼̅̅̅, 𝑂𝑇̅̅̅̅ ≅ 𝐼𝐸̅̅̅, 𝑚∠𝐻𝑂𝑇 > 𝑚∠𝑃𝐼𝐸

Prove: |𝐻𝑇| > |𝑃𝐸|

[illustration: triangles HOT and PIE with marked congruent sides and an angle at O and at I]

Proof:

[illustration: triangle HOT with point S and the bisector construction; triangle PIE with marked sides and angle]

1. Locate 𝑆 on the same side of 𝑂𝑇⃡ as 𝐻 such that ∠𝐻𝑂𝑆 ≅ ∠𝑃𝐼𝐸.
   • Consider point 𝑅 on 𝑂𝑆⃡ so that 𝑂𝑅̅̅̅̅ ≅ 𝑂𝑇̅̅̅̅ ≅ 𝐼𝐸̅̅̅
   • Consequently, ∆𝐻𝑂𝑅 ≅ ∆𝑃𝐼𝐸 by SAS Triangle Congruence Postulate.
2. Construct the bisector 𝑂𝑄̅̅̅̅ of ∠𝑇𝑂𝑅 such that 𝑄 is on 𝐻𝑇̅̅̅̅ and ∠𝑇𝑂𝑄 ≅ ∠𝑅𝑂𝑄.
   • Consequently, ∆𝑇𝑂𝑄 ≅ ∆𝑅𝑂𝑄 by SAS Triangle Congruence Postulate because 𝑂𝑄̅̅̅̅ ≅ 𝑂𝑄̅̅̅̅ by reflexive property of equality and 𝑂𝑅̅̅̅̅ ≅ ̅𝑂𝑇̅̅̅̅ from construction no.1. So, 𝑅𝑄̅̅̅̅ ≅ 𝑄𝑇̅̅̅̅ because corresponding parts of congruent triangles are congruent (CPCTC).

Statements	Reasons
1. |𝐻𝑇| = |𝐻𝑄| + |𝑄𝑇|.	1. Segment Addition Postulate
2. |𝐻𝑇| = |𝐻𝑄| + |𝑅𝑄|.	2. Substitution Property of Equality
3. In ∆𝐻𝑄𝑅, |𝐻𝑄| + |𝑅𝑄| > |𝐻𝑅|.	3. Triangle Inequality Theorem 3 (𝑆1 + 𝑆2 > 𝑆3)
4. |𝐻𝑇| > |𝐻𝑅| (Using statements 2 and 3)	4. Substitution Property of Inequality
5. |𝐻𝑅| = |𝑃𝐸|	5. Definition of congruent triangles.
6. |𝐻𝑇| > |𝑃𝐸| (Using statement in construction 1 and question number 3.)	6. Substitution Property

Converse of Hinge Theorem or SSS Inequality Theorem

We are almost finished proving the theorems on inequalities in triangles. You just proved the Hinge theorem. Let us see how the Converse of Hinge Theorem is done.

If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle in the first triangle is larger than the included angle in the second triangle.

Given: ∆𝐹𝑈𝑁 and ∆𝐴𝑅𝑇 𝐹𝑈̅̅̅̅̅ ≅ 𝐴𝑅̅̅̅̅̅, 𝑈𝑁̅̅̅̅ ≅ 𝑅𝑇̅̅̅̅̅, |𝐹𝑁| > |𝐴𝑇|

Prove: 𝑚∠𝐹𝑈𝑁 > 𝑚∠𝐴𝑅𝑇

[illustration: triangles FUN and ART with marked congruent sides]

Proof:

Let us use indirect proof to show that 𝑚∠𝐹𝑈𝑁 > 𝑚∠𝐴𝑅𝑇. Thus, we will assume that 𝑚∠𝐹𝑈𝑁 ≯ 𝑚∠𝐴𝑅𝑇.

Statements	Reasons
1. ∆𝐹𝑈𝑁 and ∆𝐴𝑅𝑇 𝐹𝑈̅̅̅̅̅ ≅ 𝐴𝑅̅̅̅̅̅, 𝑈𝑁̅̅̅̅ ≅ 𝑅𝑇̅̅̅̅̅, |𝐹𝑁| > |𝐴𝑇|	1. Given
2. ∠𝐹𝑈𝑁 ≅ ∠𝐴𝑅𝑇 or 𝑚∠𝐹𝑈𝑁 < 𝑚∠𝐴𝑅𝑇.	2. Assumption: 𝒎∠𝑭𝑼𝑵 ≯ 𝒎∠𝑨𝑹𝑻.
3. Consider: ∠𝐹𝑈𝑁 ≅ ∠𝐴𝑅𝑇. If ∠𝐹𝑈𝑁 ≅ ∠𝐴𝑅𝑇, then ∆𝐹𝑈𝑁 ≅ ∆𝐴𝑅𝑇.	3. SAS
4. 𝐹𝑁̅̅̅̅ ≅ 𝐴𝑇̅̅̅̅̅.	4. CPCTC
5. The assumption that ∠𝐹𝑈𝑁 ≅ ∠𝐴𝑅𝑇 is false.	5. 𝐹𝑁̅̅̅̅ ≅ 𝐴𝑇̅̅̅̅̅ contradicts the given that |𝐹𝑁| > |𝐴𝑇|.
6. Consider 𝑚∠𝐹𝑈𝑁 < 𝑚∠𝐴𝑅𝑇. If 𝑚∠𝐹𝑈𝑁 < 𝑚∠𝐴𝑅𝑇, then |𝐹𝑁| < |𝐴𝑇|.	6. Hinge Theorem
7. The assumption that ∠𝐹𝑈𝑁 < ∠𝐴𝑅𝑇 is False.	7. |𝐹𝑁| < |𝐴𝑇| contradicts the given that |𝐹𝑁| > |𝐴𝑇|.
8. Therefore, 𝑚∠𝐹𝑈𝑁 > 𝑚∠𝐴𝑅𝑇 must be True.	8. Assumption that 𝑚∠𝐹𝑈𝑁 ≯ 𝑚∠𝐴𝑅𝑇 is proven to be false.

Hurray! All theorems on inequalities in triangles have already been proven. You are now equipped with skills on how to write proofs in two-column form. Always remember that in writing proofs, the properties of equality and congruence will be used as bases for reasoning. Moreover, you should be equipped with the needed knowledge such as undefined terms, definitions, postulates, and theorems in geometry since these are essential to support the statement of a proof.

Having proven all the theorems on triangle inequalities, you can now take a closer look on the applications of these theorems in any aspects of real-life situations. You can use these skills in identifying unknown quantities in triangle inequalities and in justifying them. This time, let us apply what you know about inequalities to solve problems about triangles. Learn how to compare the lengths of sides and angle measures without knowing the actual measures. Here is a math problem you might have encountered in real life that will make use of one of the triangle inequality theorems. It will help you gain ideas and hints on how to justify real life problems regarding inequalities in triangles.

Illustrative Example:

Ellaine is going to the market. She can choose one of the walking routes, either the wide roads or the narrow one as illustrated. Ellaine was told by her mother to go back home as fast as possible.

1. If you were Ellaine, which road will you choose? What is your basis in choosing which road to take when the measures of the roads are not given in the illustration? Which among the theorems will support your answer?
2. If Ellaine were not told by her mother to go back home as fast as possible, do you think Ellaine would have taken the wide roads? Justify your answer.

[illustration: wide roads and a narrow road forming a triangle between Ellaine’s residence (top left) and the Market (bottom right)]

In this problem, we do not know the lengths of the roads, but we can see that the wide and the narrow roads together form a triangle. Here, let us represent the wide roads as the first two sides of the triangle and the narrow one represents the third side. If you were Ellaine, it would be better to take the narrow road. It can be justified by Triangle Inequality Theorem 3 (𝑆1 + 𝑆2 > 𝑆3), which states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. If Ellaine was not told to go back home as fast as possible, she could have taken the wider roads for more walking exercise.

What’s More

Activity 1: Complete Me!

Directions: Write the statement or reason in the two-column proof. Use the different definitions, postulates, theorems in geometry to support the given statement.

Given: 𝑇𝑃̅̅̅̅ ≅ 𝐵𝐴̅̅̅̅̅, 𝑃𝑅̅̅̅̅ ≅ ̅𝐵𝐶̅̅̅̅, |𝑇𝑅| > |𝐴𝐶|

Prove: 𝑚∠𝑇𝑃𝑅 > 𝑚∠𝐴𝐵𝐶

[illustration: triangle TPR with marked sides and triangle ABC with marked sides; labels P, B, T, R, A, C]

Proof:

Statements	Reasons
1. 𝑇𝑃̅̅̅̅ ≅ ̅𝐵𝐴̅̅̅̅̅, 𝑃𝑅̅̅̅̅ ≅ 𝐵𝐶̅̅̅̅̅, |𝑇𝑅| > |𝐴𝐶|	1. Given
2. ∠𝑇𝑃𝑅 ≅ ∠𝐴𝐵𝐶 or 𝑚∠𝑇𝑃𝑅 < 𝑚∠𝐴𝐵𝐶.	2. Assumption: 𝑚∠𝑇𝑃𝑅 ≯ 𝑚∠𝐴𝐵𝐶
3. Consider: ∠𝑇𝑃𝑅 ≅ ∠𝐴𝐵𝐶. If ∠𝑇𝑃𝑅 ≅ ∠𝐴𝐵𝐶, then ∆𝑇𝑃𝑅 ≅ ∆𝐴𝐵𝐶 .	3. _____
4. 𝑇𝑅̅̅̅̅ ≅ 𝐴𝐶̅̅̅̅̅.	4. _____
5. The assumption that ∠𝑇𝑃𝑅 ≅ ∠𝐴𝐵𝐶 is false.	5. 𝑇𝑅̅̅̅̅ ≅ 𝐴𝐶̅̅̅̅̅ contradicts the given that |𝑇𝑅| > |𝐴𝐶|.
6. Consider 𝑚∠𝑇𝑃𝑅 < 𝑚∠𝐴𝐵𝐶. If 𝑚∠𝑇𝑃𝑅 < 𝑚∠𝐵𝐴𝐶, then |𝑇𝑅| < |𝐴𝐶|.	6. _____
7. The assumption that 𝑚∠𝑇𝑃𝑅 < 𝑚∠𝐴𝐵𝐶 is _____.	7. |𝑇𝑅| < |𝐴𝐶| contradicts the given that |𝑇𝑅| > |𝐴𝐶|.
8. Therefore, 𝑚∠𝑇𝑃𝑅 > 𝑚∠𝐴𝐵𝐶 must be _____.	8. Assumption that 𝑚∠𝑇𝑃𝑅 ≯ 𝑚∠𝐴𝐵𝐶 is proven to be false.

Activity 2: Proving

Directions: Write the statements supported by the reasons on the right side of the two-column proof. Write your answer on a separate sheet of paper.

Given: 𝑀𝑂̅̅̅̅̅ ≅ 𝑂𝐸̅̅̅̅̅

Prove: |𝐸𝑅| + |𝑅𝑂| > |𝑀𝑂|

[illustration: quadrilateral-like figure with points E, R, M, O; segments ME and EO, tick marks on MO and OE]

Proof:

Statement	Reasons
1.	1. Given
2.	2. Definition of congruent segments
3. |𝐸𝑅| + |𝑅𝑂| > |𝑂𝐸|.	3.
4. |𝐸𝑅| + |𝑅𝑂| > |𝑀𝑂|.	4.

Activity 3: Reasoning

Directions: Observe and notice the congruent sides and congruent angles shown in the illustration below. Write the reasons for the statements in the two-column proof.

Given: ∆𝑇𝐸𝑀

Prove: 𝐸𝑇 + 𝐸𝑀 > 𝑇𝑀

[chart: geometric diagram; triangle with vertices T, S, M and point E; angle marks 1 and 2 at T; tick marks on segments; labels T, S, E, M]

Proof:

Statements	Reasons
1. |𝐸𝑆| = |𝐸𝑇|.	1.
2. ∆𝑇𝑆𝐸 is isosceles triangle.	2.
3. ∠𝐸𝑇𝑆 ≅ ∠𝑇𝑆𝐸.	3.
4. ∠𝑇𝑆𝐸 ≅ ∠𝑇𝑆𝑀.	4.
5. ∠𝐸𝑇𝑆 ≅ ∠𝑇𝑆𝑀.	5.
6. 𝑚∠𝑆𝑇𝑀 = 𝑚∠𝑆𝑇𝐸 + 𝑚∠𝐸𝑇𝑀.	6.
7. 𝑚∠𝑆𝑇𝑀 = 𝑚∠𝑇𝑆𝑀 + 𝑚∠𝐸𝑇𝑀.	7.
8. 𝑚∠𝑆𝑇𝑀 > 𝑚∠𝑇𝑆𝑀.	8.
9. |𝑆𝑀| > |𝑇𝑀|.	9.
10. |𝐸𝑆| + |𝐸𝑀| = |𝑆𝑀|.	10.
11. |𝐸𝑆| + |𝐸𝑀| > |𝑇𝑀|.	11.
12. |𝐸𝑇| + |𝐸𝑀| > |𝑇𝑀|.	12.

Activity 4: Justifying

Directions: Read and understand the problem and answer the questions that follow.

Two bikers Chris and Rey have uniform biking speed and are headed in opposite directions from their school. Chris is headed to the east while Rey is to the west. After biking 5 kilometers each, both of them took right turns at different angles. Chris turned at 30⁰ while Rey at 40⁰. Both continued biking and covered another 6 kilometers before taking a rest.

Questions:

1. Who is farther from the school, Chris or Rey? Justify your answer.
2. Which inequality theorem justifies your answer? Explain.

What I Have Learned

Directions: In your own words, complete each statement below based on the concepts you have learned in the topic.

1. I have learned that the different theorems of triangle inequalities are ________________________________________________________________.
2. I have learned that the theorems on inequalities in triangle or triangles can be justified through ______________________________________________.
3. I need to be equipped with the knowledge of ________________________________________________________ to support the statement of my proofs.
4. I learned that proofs on theorems on inequalities can be written in many ways, but a two-column proof is _________________________________________.
5. In writing proofs, I developed my skills in ___________________________________________________________.

What I Can Do

Directions: Observe your surroundings and make a design where triangle inequalities are applied. Make justifications that the triangle inequality exists in the created design.

Criteria	Points: 4	Points: 3	Points: 2	Points: 1	Rating
Appropriateness	The output contains a design which shows concrete example of triangle inequalities in real life.	The output contains a design which shows an example of triangle inequalities in real life.	The output contains a design which show an example of triangle inequalities but does not show its application in real life	The output contains a design which shows an example not connected to triangle inequalities.
Clarity/precision	All parts of the design are clear and precise	Some parts of the design are unclear and not precise	Most parts of the design are unclear and not precise	The drawing is unclear and not precise
Justification	Insight and depth of content understanding are evident	Some depth of content understanding are evident	Lacks content understanding	Does not give a clear understanding of the content

Assessment

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

1. Which of the following statements is true about the figure?

[chart: triangle DEA with point R on side DA; labels D, E, A, R; angle numbers 1 through 6 and 7 printed]

A. m∠6 = m∠3 B. m∠5 < m∠3 C. m∠5 > m∠1 D. m∠4 > m∠2

2. Study the figure in item 1. Notice that 𝑚∠5 > 𝑚∠3 and 𝑚∠5 > 𝑚∠1. Which theorem justifies these observations?

   
   

   A. Exterior Angle Inequality Theorem
   B. Triangle Inequality Theorem 1 (Ss→Aa)
   C. Triangle Inequality Theorem 2 (Aa→Ss)
   D. Triangle Inequality Theorem 3 (S1 + S2 > S3)

   
   

3. In ∆𝑃𝐴𝑌, if |𝑃𝐴| = 4 𝑐𝑚, |𝐴𝑌| = 5 𝑐𝑚, and |𝑃𝑌| = 3 𝑐𝑚, which statement is true?

   
   

   A. 𝑚∠𝐴 > 𝑚∠𝑌. C. 𝑚∠𝐴 > 𝑚∠𝑃.
   B. 𝑚∠𝑌 > 𝑚∠𝑃. D. 𝑚∠𝑃 > 𝑚∠𝐴.

   
   

4. Which theorem justifies the statement/answer in item 3?

   
   

   A. Exterior Angle Inequality Theorem
   B. Triangle Inequality Theorem 1 (Ss→Aa)
   C. Triangle Inequality Theorem 2 (Aa→Ss)
   D. Triangle Inequality Theorem 3 (S1 + S2 > S3)

   
   

5. In ∆𝐺𝑂𝐷, |𝐺𝑂| = |𝐷𝑂| and |𝐺𝐷| > |𝐷𝑂|. Which of the following statements is NOT true?

   
   

   A. 𝑚∠𝐺𝑂𝐷 > 𝑚∠𝑂𝐷𝐺 C. 𝑚∠𝑂𝐷𝐺 = 𝑚∠𝐷𝐺𝑂
   B. 𝑚∠𝐺𝑂𝐷 > 𝑚∠𝐷𝐺𝑂 D. 𝑚∠𝐺𝑂𝐷 < 𝑚∠𝑂𝐷𝐺

   
   

6. Using the lengths of the sides shown in the triangles below, what conclusion can be formed using the Converse of Hinge Theorem?

   
   

[chart: two triangles sharing vertex P, with vertices A, T, H, P; side lengths printed: 12, 9, 12, 7]

A. |P𝑇| = |𝑃𝑇|. B. |P𝐴| = |𝑃𝐻|. C. m∠APT > m∠HPT. D. m∠HPT > m∠APT.

7. In the figure below, Khen concluded that 𝑚∠𝑇𝑃𝐴 > 𝑚∠𝑃𝑇𝐴. Which theorem on inequalities in triangle justifies Khen’s conclusion?

[chart: triangle with vertices A, P, T; tick marks on sides AP and AT]

A. Exterior Angle Inequality Theorem B. Triangle Inequality Theorem 1 (Ss→Aa) C. Triangle Inequality Theorem 2 (Aa→Ss) D. Triangle Inequality Theorem 3 (S1 + S2 >S3)

8. Joan analyzed the triangles in the given figure at the right. Which of the following is a correct observation?

[chart: quadrilateral-shaped figure with vertices F, I, E, N and diagonal FN; labels F, I, E, N; lengths printed: 9, 10, 5, 5]

A. |𝐸𝐹| = |𝐼𝐹|. B. |𝐼𝑁| > |𝐸𝑁|. C. |𝐹𝑁| < |𝐹𝑁|. D. 𝑚∠𝐸𝑁𝐹 > 𝑚∠𝐼𝑁𝐹

9. Which theorem justifies Joan’s conclusion in item no. 8?

   
   

   A. Hinge Theorem
   B. Converse of Hinge Theorem
   C. Triangle Inequality Theorem 1 (Ss→Aa)
   D. Triangle Inequality Theorem 3 (S1 + S2 > S3)

   
   

10. Using the given figure below, can you conclude that |𝐻𝑆| > |𝐻𝐹| if one of the following statements is not established: 𝐼𝐻 ≅ 𝐼𝐻, 𝐼𝐹 ≅ 𝐼𝑆, 𝑚∠𝑆𝐼𝐻 > 𝑚∠𝐹𝐼𝐻?

    
    

[chart: triangle with vertices I, F, S and point H inside; labels I, F, H, S; angle values 38° and 40° printed]

A. Yes, I will. B. No, I will not. C. It is impossible to decide. D. It depends on which statement is left out.

11. In the figure below, Chris has proved that |𝐼𝑆| > |𝐼𝑊|. Which of the following statements is NOT part of his proof?

[chart: figure with vertices S, E, W, I; segment labels 10 and 10 printed; angle 75° printed at E]

A. 𝐸𝑆 ≅ 𝐸𝑊. B. 𝐸𝐼 ≅ 𝐸𝐼. C. 𝑚∠𝑊 < 𝑚∠𝑆. D. 𝑚∠𝑊𝐸𝐼 + 𝑚∠𝑆𝐸𝐼 = 180⁰.

12. What theorem did Chris use to justify his statement/answer in item 11?

    
    

    A. Hinge Theorem
    B. Converse of Hinge Theorem
    C. Triangle Inequality Theorem 1 (Ss→Aa)
    D. Triangle Inequality Theorem 3 (S1 + S2 > S3)

    
    

13. Complete the statement: Even without actual measurements, inequalities in triangles can be justified ________ using theorems on inequalities in triangles.

    
    

    A. inductively C. specifically
    B. deductively D. generally

    
    

14. Based on the figure below, what should be the reason if the statement is 𝑚∠𝑆𝐴𝑁 = 𝑚∠1 + 𝑚∠3?

    
    

[chart: triangle SAN with point D on side SN; angle labels 1, 2, 3 printed]

A. Linear Pair Postulate
B. Substitution Property
C. Property of Inequality
D. Angle Addition Postulate

15. After a home study session, Khen and Khai went home heading different directions from Chris’ house. Khen to the east and Khai to the west. After both have walked 30 meters, Khen turned right at 70⁰ and Khai to her right at 50⁰. After another 40-meter walk, they both arrived at their respective homes. Who is farther from Chris’ house? What theorem supports your answer?

    
    

    A. Khai, Hinge Theorem
    B. Khen, Converse of Hinge Theorem
    C. Khen, Triangle Inequality Theorem 1 (Ss→Aa)
    D. Khai, Triangle Inequality Theorem 3 (S1 + S2 > S3)

    
    

Additional Activities

Directions: Observe and notice the congruent sides and angles shown in the illustration below. Write the missing statement or reason in the two-column proof.

Given: 𝐸 is the midpoint of 𝑅𝐴, ∠1 ≅ ∠2, 𝑚∠3 > 𝑚∠4

Prove: |𝑀𝐴| > |𝐷𝑅|

[chart: geometric figure with points D, M, R, E, A; angle labels 1, 2, 3, 4 printed]

Proof:

Statements	Reasons
1. ∠1≅∠2	1. Given
2. _____ is an isosceles triangle.	2. Definition of isosceles triangle
3. 𝐷𝐸 ≅ 𝑀𝐸.	3. Legs of isosceles triangle are congruent.
4. E is the midpoint of 𝑅𝐴.	4.
5.	5. Definition of a midpoint
6. 𝑚∠3 > 𝑚∠4.	6.
7. |𝑀𝐴| > |𝐷𝑅|.	7.

Answer Key

[transcriber’s note: this page is printed rotated 180 degrees; answers below were read from the rotated print, and entries marked [?] or [illegible] could not be read with certainty]

What I Know

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

Assessment

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

What’s In

I. 1. E II. 1. C I. 2. E [?] II. 2. A I. 3. F II. 3. B I. 4. A II. 4. F I. 5. B II. 5. C

What’s New

Activity 1

1. PQ ≅ PS
2. ΔPQS is isosceles triangle
3. ∠1 ≅ ∠2
4. Angle Addition Postulate
5. Property of Inequality
6. Substitution Property
7. [illegible]
8. [illegible]
9. [illegible]
10. [illegible]
11. ∠2 > ∠R [?]
12. ∠PQR > ∠PRQ [?]

Activity 2

1. [illegible]
2. [illegible]
3. ∠PQR ≅ ∠PRQ [?]
4. [illegible]
5. R: Triangle Inequality Theorem 1
6. False
7. True

Activity 3

1. PR ≅ PS
2. ΔPRS is isosceles
3. ∠PSR ≅ ∠PRS
4. Reflexive Property of Equality
5. ∠QSR ≅ ∠PRS
6. Angle Addition Postulate
7. Substitution Property
8. [illegible]
9. Triangle Inequality Theorem 2
10. [illegible]
11. PQ + PS > QR
12. PQ + PR > QR

Activity 4

1. [illegible]
2. ∠3 ≅ ∠4
3. ΔPTQ ≅ ΔRTU [?]
4. ∠QPR ≅ ∠1 [?]
5. ∠1 + ∠2 [?]
6. [illegible]
7. [illegible]

Activity 5

1. Segment Addition Postulate
2. Substitution Property of Equality
3. HQ + RQ > HR
4. Substitution Property of Equality
5. Property of Inequality

Activity 6

1. m∠FUN > m∠ART [?]
2. SAS
3. [illegible]
4. [illegible]
5. Hinge Theorem
6. False
7. True

What’s More

Activity 1: Completing

1. [illegible]
2. [illegible]
3. SAS
4. CPCTC
5. [illegible]
6. Hinge Theorem
7. False
8. True

Activity 2: Proving

1. MO ≅ OE
2. MO = DE
3. Triangle Inequality Theorem 3 (S1 + S2 > S3)
4. Substitution Property

Activity 3: Reasoning

1. By construction
2. Definition of isosceles triangle
3. Base angles of isosceles triangle are congruent
4. Reflexive Property of Equality
5. Transitive Property of Equality
6. Angle Addition Postulate
7. Substitution Property
8. Property of Inequality
9. Triangle Inequality Theorem 2
10. Segment Addition Postulate
11. Substitution Property
12. Substitution Property

Activity 4: Justifying

1. Chris
2. Hinge Theorem

What I Have Learned

Answer may vary.

What I Can Do

Answers may vary.

Additional Activities

1. [illegible]
2. ΔDEM
3. [illegible]
4. Given
5. RE = EA
6. Given
7. Hinge Theorem

References

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

Bernabe, Julieta G., et.al. (2009). Revised Edition Geometry, pp. 88-90, Gregoria Araneta Avenue, Quezon City, Philippines. SD Publications, Inc.

Gabriel, J.L., Mallari, M.T.G. Mathematics for Grade 8. Quezon City, Philippines: Educational Resources Corporation, 2017.

Orines, Fernando B., et.al. (2008). Second Edition Next Century Mathematics, pp. 201-203, Quezon Avenue, Quezon City, Philippines. Phoenix Publishing House, Inc.

Links

www.mathsisfun.com › geometry › triangles-inequalities

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