Mathematics 8 Quarter 4 Module 1: Illustrating Theorems on Triangle Inequalities

Mathematics

Quarter 4 – Module 1 Illustrating Theorems on Triangle Inequalities

[illustration: learners studying outdoors and at home, with a teacher, a globe, and a lamp]

CO_Q4_Mathematics 8_ Module 1

Mathematics – Grade 8 Alternative Delivery Mode Quarter 4 – Module 1 Illustrating Theorems on Triangle Inequalities First Edition, 2020

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Printed in the Philippines___________________________

Development Team of the Module

Writer: Melody C. Gapa

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Layout Evaluator: Devina P. Malinao

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8

Mathematics

Quarter 4 – Module 1 Illustrating Theorems on Triangle Inequalities

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.

1 CO_Q4_Mathematics 8_ Module 1

What I Need to Know

This module will assist you learn the concept of inequalities in triangles. It focuses on how to illustrate and apply theorems on triangle inequalities. You are provided with varied tasks and activities to deepen your understanding of the lesson. This module was designed to be self-sufficient for the current learning situations. The lesson is arranged to follow the standard sequence of the course in the curriculum guide. However, the order in which you read them can be changed to correspond with the textbook you are now using.

This module contains lesson on illustrating theorems on triangle inequalities:

• Exterior Angle Inequality Theorem;
• Triangle Inequality Theorem; and
• Hinge Theorem.

After going through this module, you are expected to:

1. investigate the relationship between the longest side and the largest angle in the triangle and vice versa;
2. investigate the relationship between the sum of any two sides and the remaining sides in a triangle;
3. illustrate theorems on triangle inequalities such as the Exterior Angle Inequality Theorem, Triangle Inequality Theorem, and Hinge Theorem with its converse; and
4. connect theorems in triangle inequalities in real-life setting.

2 CO_Q4_Mathematics 8_ Module 1

What I Know

Pre- Assessment

Directions: Choose the letter of the correct answer. Write your answer on a separate sheet of paper.

1. Which theorem states that if one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side?
   a. Triangle Inequality Theorem
   b. Angle-Side Relationship Theorem
   c. Exterior Angle Inequality Theorem
   d. Hinge Theorem or SAS Inequality Theorem

   
   

2. In Triangle Inequality Theorem, which of the following describes the relationship of the sum of its two sides to its third length?
   a. greater than
   b. greater than or equal to
   c. less than
   d. less than or equal to

   
   

3. Which of the following theorems on triangle inequalities states that the measure of an exterior angle of a triangle is greater than the measure of either remote interior angle?
   a. Triangle Inequality Theorem
   b. Angle-Side Relationship Theorem
   c. Exterior Angle Inequality Theorem
   d. Hinge Theorem or SAS Inequality Theorem

   
   

4. Which of the following theorems deals with the two triangles whose two corresponding sides are congruent and whose included angle are unequal?
   a. SAS Inequality Theorem
   b. SSS Inequality Theorem
   c. Angle-Side Relationship Theorem
   d. Exterior Nagle Inequality Theorem

   
   

5. Which of the following could NOT be used as the length of the sides of a triangle?
   a. 3, 3, 3
   b. 4, 5, 10
   c. 5, 7, 9
   d. 10, 20, 20

   
   

3 CO_Q4_Mathematics 8_ Module 1

6. Which of the following statements is FALSE?
   a. Any lengths of the sides can be used to form a triangle.
   b. The largest angle of a triangle is opposite to the longest side.
   c. The smallest side of a triangle is opposite to the smallest angle.
   d. The sum of any two sides of a triangle is greater than the third side.

   
   

7. 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 of the first triangle is larger than the included angle of the second. Which of the following theorems pertains to two triangles whose two corresponding sides are congruent and whose third sides are unequal?
   a. Triangle Inequality Theorem
   b. Exterior Angle Inequality Theorem
   c. Hinge Theorem or SAS Inequality Theorem
   d. Converse of Hinge Theorem or SSS Inequality Theorem

   
   

In items 8 to 10, refer to the figure at the right.

[chart: diagram; triangle TWO for items 8 to 10. Labels: W (top vertex), 91° (angle at W), T (left vertex), 42° (angle at T), O (right vertex).]

8. Which side of ∆TWO is the longest?
   a. $\overline{WT}$
   b. $\overline{OW}$
   c. $\overline{OT}$
   d. Cannot be determined

   
   

9. Which side is the shortest?
   a. $\overline{WT}$
   b. $\overline{OW}$
   c. $\overline{OT}$
   d. Cannot be determined

   
   

10. What theorem of triangle inequalities is applied?
    a. SSS Inequality Theorem
    b. Triangle Inequality Theorem
    c. Angle-Side Relationship Theorem
    d. Exterior Angle Inequality Theorem

    
    

11. In ∆ $NET$, $\overline{NE}$ = 14 cm, $\overline{ET}$ = 15 cm, $\overline{NT}$ = 16 cm. What is the correct order of the angles when arranged from least to greatest?
    a. ∠E, ∠N, ∠T
    b. ∠E, ∠T, ∠N
    c. ∠T, ∠E, ∠N
    d. ∠T, ∠N, ∠E

    
    

4 CO_Q4_Mathematics 8_ Module 1

12. Given the figure at the right, which of the following angles is an exterior angle of ∆ $DER$?
    a. ∠3
    b. ∠4
    c. ∠5
    d. ∠6

[chart: diagram; triangle DER with point R on side DA and point A, with angles numbered 1 to 6. Labels: D, E, R, A, and angle numbers 1, 2, 3, 4, 5, 6.]

13. Which of the following statement about the given figure is true?
    a. ∠KIN = ∠DNI
    b. ∠KIN > ∠IND
    c. ∠DNI > ∠KIN
    d. ∠IKN > ∠IND

[chart: diagram; triangle IKN with side KN extended through N to D. Labels: I (top vertex), K (bottom left), N (bottom right), D (on extension beyond N).]

14. In ∆ FRY, which of the following shows the correct arrangement of its sides in descending order?
    a. $\overline{FY}$, $\overline{FR}$, $\overline{RY}$
    b. $\overline{FR}$, $\overline{RY}$, $\overline{FY}$
    c. $\overline{FY}$, $\overline{RY}$, $\overline{FR}$
    d. $\overline{RY}$, $\overline{FR}$, $\overline{FY}$

[chart: diagram; triangle FRY. Labels: R (top vertex), 88° (angle at R), F (bottom left), 54° (angle at F), Y (bottom right), 38° (angle at Y).]

15. Ruel, Geralyn, Reymond and Shiela were given a 24-inch piece of stick each. They were instructed to create a triangle. Each cut the stick in their own chosen lengths as follows: Ruel—8 in, 8 in, 8 in; Geralyn—5 in, 7 in, 12 in; Reymond—9 in, 7 in, 8 in; and Shiela—6 in, 8 in, 10 in. Who among them could NOT be able to form a triangle?
    a. Ruel
    b. Geralyn
    c. Shiela
    d. Reymond

Lesson 1: Illustrate Theorems on Triangle Inequalities

Have you ever thought how the engineers and architects make use of triangular concept in designing a house? What mathematical concept may justify the triangular designs that they use? The answers to these questions are revealed in this module.

What’s In

Investigate Me!

Directions: Use the figure below to answer the questions that follow. Write your answer on a separate sheet of paper.

1. What is the included side in ∠B and ∠C? in ∠E and ∠F?
2. What is the included angle in 𝐴𝐶̅̅̅̅ and 𝐵𝐶̅̅̅? in 𝐷𝐹̅̅̅̅ and 𝐸𝐹̅̅̅̅?
3. What is the sum of the interior angles of ∆ABC? ∆DEF?
4. If ∠B ≅ ∠E, and ∠C ≅ ∠F, what additional information is required to tell that the triangles are congruent using SAS Congruence?
5. If 𝐴𝐶̅̅̅̅ ≅ 𝐷𝐹̅̅̅̅ and 𝐵𝐶̅̅̅ ≅ 𝐸𝐹̅̅̅̅, what additional information is required to tell that the triangles are congruent using SSS Congruence?

[illustration: two triangles, ABC on the left and DEF on the right, with vertices B, F, D, A, C, E labeled; angle arcs and tick marks show the congruent parts]

Questions:

1. Did you find any difficulty in the conduct of the activity?
2. What did you do to overcome this difficulty?

What’s New

The activity Investigate Me is a recall of your understanding on the different angles and sides of triangles. In the next activity you will be using some measuring tools, pen and paper to further understand the concept behind the previous activity.

Measuring Angles and Sides

Materials needed: protractor, ruler, paper and pencil/pen

Directions: Using measuring tools, construct an angle based on the given measurements of the angle and the length of the sides. Lastly connect the end point of each side to form a triangle and find its measurement. Write your answer on a separate sheet of paper. An illustration is provided to guide you in how to do the measurement.

[illustration: triangle with Side 1 along the base, Side 2 along the left slanted edge, and Angle at the bottom left vertex]

Labels: Side 2, Angle, Side 1

1. Given that Side 1 is 3 inches and Side 2 is 2 inches. Find the length between the end points of the side when the angle measures:

Angle:	30°	45°	90°	120°	150°
Length between the end points

2. Given that both sides of the angle measures 2 inches. Find the length between the end points of the side when the angle measures:

Angle:	30°	45°	90°	120°	150°
Length between the end points

Questions:

1. What can you say about the length of the side connecting the endpoints? What pattern were you able to establish?
2. What can you say about the relationship of the measurement of the angle and the length of the opposite side?
3. Is there a relationship between the length of a side of a triangle and the measure of the angle opposite it? If your answer is YES, indicate its relationship.

Extending the triangle

Materials needed: protractor, ruler, paper and pencil/pen

Directions: Using measuring tools extend side 1 of the triangles formed in the activity Measuring Angles and Sides as illustrated below. Supply the measurements of the angle 2 and angle 3 in each table. Write your answer on a separate sheet

[illustration: triangle with Side 2 on the left slanted edge, Side 1 along the base extended to the right, Angle 1 at the bottom left vertex, Angle 2 at the apex, and Angle 3 as the exterior angle at the bottom right vertex]

Labels: Side 2, Angle 2, Angle 1, Angle 3, Side 1

1. Given that Side 1 is 2 inches and Side 2 is 3 inches. Find measurement of angles 2 and 3 when the measurement of angle 1 is:

Angle 1:	30°	45°	90°	120°	180°
Angle 2:
Angle 3:

2. Given that both sides of the angle measures 2 inches. Find measurement of angles 2 and 3 when the measurement of angle 1 is:

Angle 1:	30°	45°	90°	120°	180°
Angle 2:
Angle 3:

Questions:

1. Is there a relationship between the measurement of angle 1 and angle 2? How about between angle 1 and angle 3? Is the relationship also true between angle 2 and angle 3?
2. What generalization can you draw from the measurement of angle 3 to the measurement of angle 1 and measurement of angle 2?

What is It

There are various theorems on triangle inequalities. These theorems can be illustrated in one triangle and in two triangles. The most commonly used theorems in one triangle are Angle-Side Relationship Theorem, Triangle Inequality Theorem, and Exterior Angle Inequality Theorem. On the other hand, the Hinge Theorem (SAS Inequality Theorem) and the Converse of Hinge Theorem (SSS Inequality Theorem) illustrate the inequalities in two triangles. To fully understand these theorems, read and understand the figures and illustrations below.

Inequalities in One Triangle

In your activity in Measuring Angles and Sides you found out that there is a relationship between the measure of the angle and the length of the side of a triangle. This theorem is referred to as Angle-Side Relationship Theorem which states that: In a triangle, the side opposite the larger angle is the longer side and vice versa. To know more about this theorem, examine the following examples:

Example 1: Compare the length of the sides of the following triangle.

[illustration: triangle with vertices U (top), F (bottom left), and N (bottom right); angle 105° at U and angle 50° at F]

Solution:

Step 1: Find the measure of the third angle. The sum of all the angles in any triangle is 180°.

∠F + ∠U + ∠N = 180° — Sum of interior angles of a triangle 50° + 105° + ∠N = 180° — By substitution ∠N + 155° = 180° — Adding 50° and 105° ∠N + 155° + (−155°) = 180° + (−155°) — Addition property of equality ∠N + 0 = 180° − 155° — Identity property of addition ∠N = 25° — By simplifying

Step 2: Look at the relative sizes of the angles and compare.

∠N < ∠F < ∠U

Step 3: Following the angle-side relationship we can order the sides accordingly. Remember it is the side opposite the angle.

𝐹𝑈̅̅̅̅ < 𝑈𝑁̅̅̅̅ < 𝐹𝑁̅̅̅̅ — Thus, 𝐹𝑁̅̅̅̅ is the longest side since it is the opposite side of the largest angle, ∠U, while 𝐹𝑈̅̅̅̅ is the shortest side whose opposite angle, ∠N measures 25°.

Example 2: Compare the measure of the angles of the following triangle.

[illustration: triangle with vertices A (top), D (bottom left), and Y (bottom right); side DA labeled 5, side AY labeled 3, side DY labeled 6]

Solution:

Step 1: Since the length of the sides were given, we can easily compare the lengths from shortest to longest.

𝐴𝑌̅̅̅̅ < 𝐷𝐴̅̅̅̅ < 𝐷𝑌̅̅̅̅

Step 2: Following the angle-side relationship we can order the angles opposite to these sides accordingly.

∠𝐷 < ∠𝑌 < ∠𝐴

Therefore, ∠𝐴 is the largest angle whose is opposite to side DY whose side is 6 units. It follows with the other angles. Example 1 and Example 2 illustrate the angle-side relationship theorem.

Angle-Side Relationship Theorem

If two angles of a triangle are not congruent, then the larger side is opposite the larger angle.

If two sides of a triangle are not congruent, then the larger angle is opposite the larger side.

How can we form triangles?

Not any three lengths can form the sides of a triangle. Let a, b, and c be the lengths of the sides of a triangle. A triangle can only be formed under the following conditions:

a + b > c a + c > b b + c > a

Example 3: In ∆ FIT, find the range of the possible lengths of 𝐹𝑇̅̅̅̅.

[illustration: triangle with vertices I (top), F (bottom left), and T (bottom right); side FI labeled 5, side IT labeled 8, side FT labeled ?]

(Note: The figure is not drawn to scale.)

Solution: Let a = 5 and b = 8

Step 1: Using the triangle inequality theorem for the above triangle gives us three statements:

a + b > c ⟹ 5 + 8 > c ⟹ c < 13 a + c > b ⟹ 5 + c > 8 ⟹ c > 3 b + c > a ⟹ 8 + c > 5 ⟹ c > -3 (disregard because lengths must be positive)

Step 2: Combining the two valid statements:

3 < c < 13 (The length of 𝐹𝑇̅̅̅̅ is greater than 3 and less than 13)

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the remaining side.

Let a, b, and c be the lengths of a triangle. These lengths may only form a triangle if the three conditions are satisfied:

𝑎 + 𝑏 > 𝑐; 𝑎 + 𝑐 > 𝑏; 𝑐 + 𝑏 > 𝑎.

Let b an unknown side of a triangle. To find the range of possible measure of side b, the inequality below may be used:

(a- c) < b < (a + c)

Example 4: Which length/s can form a triangle?

A.) 3, 4, 6 B.) 5, 6, 11 C.) 2, 3, 9

Solution: Check the lengths given if it will form a triangle using Triangle Inequality Theorem.

Given lengths	a = 3, b = 4, c = 6	a = 5, b = 6, c = 11	a = 2, b = 3, c = 9
a + b > c	3 + 4 > 6 True	5 + 6 > 11 False	2 + 3 > 9 False
a + c > b	3 + 6 > 4 True	5 + 11 > 6 True	2 + 9 > 3 True
b + c > a	4 + 6 > 3 True	6 + 11 > 5 True	3 + 9 > 2 True
Decision	Triangle	Not a Triangle	Not a Triangle

Based on the Triangle Inequality Theorem, only option A satisfies the condition that will form a triangle.

Alternative Solution: Using a broomstick or a barbecue stick, cut them to each desired length and try to form them into triangles.

As we form them, these are the figures formed:

A.) 3, 4, 6 B.) 5, 6, 11 C.) 2, 3, 9

[photo: three sticks arranged as a triangle, a single straight stick, and two sticks forming an open figure]

Based on the figures, only option A formed a triangle. Option B formed a straight line instead of a triangle and option C is short of sides that it cannot form a triangle. If we compare the sum of the two sides of options A, B, and C, only option A satisfied the conditions of Triangle Inequality Theorem.

Exterior Angle Inequality Theorem

The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

[chart: diagram; Exterior Angle Inequality Theorem]

Labels: O (top vertex), L (bottom left vertex), V (bottom right vertex), E (point on the extension of the base beyond V); Remote Interior Angles (the angles at L and O); Exterior Angle (the angle at V between side OV and segment VE)

Example 5: In the figure above, if m∠L= 45° and m∠O= 105°, then by Exterior Angle Inequality Theorem:

𝑚∠𝑂𝑉𝐸 > 𝑚∠𝐿 or 𝑚∠𝑂𝑉𝐸 > 𝑚∠𝑂 𝑚∠𝑂𝑉𝐸 > 45° or 𝑚∠𝑂𝑉𝐸 > 105°

Inequalities in Two Triangle

The previous theorems deal with inequalities in one triangle, this time, we will be dealing with inequalities in two triangles.

Example 6: Compare the lengths of the third side of ∆ BLU and ∆ RED.

[illustration: triangle BLU with vertices L (top), B (bottom left), U (bottom right), angle 100° at L; triangle RED with vertices E (top), R (bottom left), D (bottom right), angle 91° at E; tick marks show BL ≅ RE and LU ≅ ED]

(Note: The figure is not drawn to scale.)

Given: 𝐵𝐿̅̅̅̅ ≅ 𝑅𝐸̅̅̅̅ — Solution: ∠L > ∠E 𝐿𝑈̅̅̅̅ ≅ 𝐸𝐷̅̅̅̅ — 100° > 91° 𝐵𝑈̅̅̅̅ ? 𝑅𝐷̅̅̅̅ — 𝐵𝑈̅̅̅̅ > 𝑅𝐷̅̅̅̅

Based on the angle-side theorem, the opposite side of the greater angle is longer side.

From this example, the corresponding two sides of two triangles are congruent but the measure of their included angles differ, it follows that the opposite side of the greater angle is longer. Thus, the opposite side of ∠L which is 𝐵𝑈̅̅̅̅ is greater than the opposite side of ∠E which is 𝑅𝐷̅̅̅̅. This theorem is what we called the Hinge Theorem or the SAS Inequality Theorem.

Hinge Theorem or SAS Inequality Theorem

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

Are you familiar with hinge devices?

[photo: a metal door hinge]

Hinge devices are used to fasten or join two things together and allow adjustment, rotation, twisting or pivoting. The picture in the left is called hinge which is attached to our doors at home. It is responsible for the opening and closing of our doors. Scissors, compass and folding ladder are just some of the examples of hinge devices.

Example 7: Which of the following pictures show a wide opening of the scissors?

[illustration: two photos of a pair of scissors opened at different angles, with dashed lines showing the opening; labeled A (left) and B (right)]

A B

Answer: Obviously, the answer is Picture B since its angle is greater than that of Picture A. Although, a pair of scissors has the same length, its opening (dashed line) depends on the angle of adjustment of the user.

Conversely, we can also tell which included angle is greater based on the length of the third side given that corresponding two sides of two triangles are congruent. This theorem is the converse of Hinge Theorem or the SSS Inequality Theorem.

Converse of Hinge Theorem or SSS Inequality Theorem

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 of the first triangle is larger than the included angle of the second.

Example 8: Given that ∆SAY and ∆TEL have two congruent sides as shown in the figure, which angle is greater, ∠A or ∠E?

[illustration: two triangles. Triangle SAY has vertices S (bottom left), A (top), Y (bottom right). Triangle TEL has vertices E (top), T (middle left), L (bottom right). Tick marks show congruent sides.]

Answer:

In the figure, we can observe that $\overline{SA} \cong \overline{TE}$ and $\overline{AY} \cong \overline{EL}$. It is also given that $\overline{SY}$ is longer than $\overline{TL}$. With these, the Converse of Hinge Theorem or SSS Inequality Theorem tells us that ∠A is greater than ∠E.

In summary, the following are theorems in triangle inequalities.

INEQUALITIES	THEOREM	IF-THEN STATEMENT
Inequalities in One Triangle	Angle-Side Relationship	If two angles of a triangle are not congruent, then the larger side is opposite the larger angle.
		If two sides of a triangle are not congruent, then the larger angle is opposite the larger side.
	Triangle Inequality Theorem	The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
	Exterior Angle Inequality Theorem	The measure of an exterior angle of a triangle is greater than the measure of its either remote interior angle.
Inequalities in Two Triangles	Hinge Theorem or SAS Inequality Theorem	If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second.
	Converse of Hinge Theorem or SSS Inequality Theorem	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 of the first triangle is larger than the included angle of the second.

What’s More

Activity 1: Identify Me

Directions: Write your answer on a separate sheet of paper. Answer what is asked in the given problem.

[illustration: triangle ABC with vertex B at top, A at bottom left, C at bottom right; angles labeled 70° at B, 65° at A, 45° at C]

a. What is the longest side in a ∆ABC? How about the shortest side? b. What theorem did you apply in determining the longest and shortest side?

[illustration: triangle SAY with vertex S at top, A at bottom left, Y at bottom right; side labels 11 (S to A), 12 (S to Y), 13 (A to Y)]

a. What is the smallest angle of ∆SAY? How about the largest angle? b. What theorem did you apply in determining the smallest and largest angle?

[illustration: triangle VOL with vertex V at left, O at right, and E at top; angle labels 123° at E-side and 30° at V; point L beyond O]

a. What is the measurement of ∠LOV? b. Using the relation symbols >, <, or = , complete the statements below: m∠LOV ___ m∠E and m∠LOV____ m∠V. c. What theorem did you use to justify the statements above?

[illustration: two triangles. Left triangle has vertices X, Y, Z with tick marks on sides. Right triangle has vertices A, B, C with tick marks on sides.]

a. Using the relation symbols >, <, or = , complete the statement below: m∠Z___ m∠C b. What theorem did you use to justify the statement above?

[illustration: two triangles. Left triangle has vertices N, W, O with tick marks and angle marks. Right triangle has vertices S, A, E with tick marks and angle marks.]

a. Using the relation symbols >, <, or = , complete the statement below: $\overline{NW}$ ___ $\overline{AS}$ b. What theorem did you use to justify the statement above?

Activity 2: Am I a Triangle?

Directions: Which of the following could be the lengths of the sides of a triangle? Put a triangle ( ) if it forms a triangle and (X) if it does not form a triangle.

[transcriber’s note: a triangle symbol is drawn over the word “forms” in the printed directions; it is shown here as the printed parentheses ( )]

_______1. 1, 2, 3 ________4. 4, 8, 11 _______2. 17, 16, 9 ________5. 5, 13, 6 _______3. 9, 11, 18

Activity 3: Make It True

Directions: Modified True or False. Write True if the statement is correct, but if it’s False, change the underlined words to make it right.

__________1. The measure of an exterior angle of a triangle is less than the measure of either remote interior angle.

__________2. If one side of a triangle is longer than a second, then the angle opposite the first side is larger than the angle opposite the second side.

__________3. The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

__________4. If one angle of a triangle is larger than a second angle, then the side adjacent the first angle is longer than the side adjacent the second angle.

__________5. If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second.

What I Have Learned

Fill in the blanks!

Directions: Complete the paragraph below by filling in the blanks with correct word/s or figure/s which you can choose from the box below. Each word or figure may be used repeatedly. Write your answer on a separate sheet.

Word box
longer than	included angle	greater than	Hinge Theorem
Triangle Inequality Theorem	Angle-Side Relationship	Exterior Angle Inequality	SSS Inequality Theorem
remote interior angle	opposite	inequalities	remote exterior angle

There are two types of theorems of inequalities in triangles. One is the Inequalities in One Triangle. One theorem under the inequalities in one triangle is the ________________ in which if two angles of a triangle are not congruent, then the larger side is __________ to the larger angle. The second one is the ____________________ which states that the sum of the lengths of any two sides of a triangle is ____________ the length of the third side. The third one is the ____________________ Theorem which states that the measure of an exterior angle of a triangle is greater than the measure of its ____________________.

On the other hand, there are also theorems of the inequalities in Two Triangles. One of its theorem is the _______________ or SAS Inequality Theorem in which if the two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle in the second, then the third side of the first triangle is _____________ the third side of the second. The other theorem is the Converse if Hinge Theorem or ______________________ which states that 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 _____________ of the first triangle is larger than the included angle of the second.

What I Can Do

Activity 1

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

In the map below, the locations form a triangle. Dina, a student, is at home while Alvin, her classmate is at school. In this situation, answer the questions that follow.

[chart: map diagram showing a triangle formed by Home, School, and Church, with Market on the same line as Home and Church; angles labeled 80° at School, 60° at Home, 40° at School-to-Church side near Church, and 140° at Church; a north arrow labeled N at upper right]

Questions:

1. Who is farther from Church given their locations, Dina or Alvin? What triangle inequality theorem is applied in this situation?
2. Dina requested Alvin to pick her up in going to Church. Will Alvin be able to travel the shortest path to go to Church given Dina’s request? What triangle inequality theorem is applied in this situation?
3. Given the same time started and same rate of walking, who among the two can arrive at the Market first?
4. If a new road will be constructed connecting the School and the Market, is the constructed road longer than the road between Home and Church? What triangle inequality theorem is applied in this situation?

Activity 2

Directions: Hinges are used to fasten two things together and allow adjustment, rotation, twisting, or pivoting. Your task is to identify and list down hinge devices you know and determine its function and its relation to Hinge Theorem. You can list hinge devices as many as you can but not less than 3. Write your answer on a separate sheet of paper. For your guidance, you may use the template on the next page.

Hinge Devices	Function	Relationship to Hinge Theorem

For rating your output, please refer to the rubric provided.

	4	3	2	1
Content	The discussion of the function of hinge devices listed is correct, clear, and realistic and the relationship to Hinge Theorem is properly illustrated.	The discussion of the function of hinge devices listed is correct, clear, and realistic and the relationship to Hinge Theorem is not properly illustrated.	The discussion of the function of hinge devices listed is correct, clear, and realistic but did not indicate the relationship to Hinge Theorem.	The discussion of the function of hinge devices listed is not clear and did not indicate the relationship to Hinge Theorem.
Quantity	Listed more than 3 hinge devices with complete function and discussion to its relation to Hinge Theorem.	Listed only 3 hinge devices with complete function and discussion to its relation to Hinge Theorem.	Listed only 2 hinge devices with complete function and discussion to its relation to Hinge Theorem.	Listed only 1 hinge devices with complete function and discussion to its relation to Hinge Theorem.

Assessment

Directions: Read the questions carefully. Write the letter that corresponds to your answer on a separate sheet of paper.

1. Which of the following statements is true based on Angle-Side Relationship Theorem?
   A. The largest side of a triangle is opposite with the smallest angle.
   B. The largest side of a triangle is adjacent with the smallest angle.
   C. The smallest side of a triangle is opposite with the smallest angle.
   D. The smallest side of a triangle is adjacent with the smallest angle.

   
   

2. The sum of the lengths of any two sides of a triangle is always _________.
   A. half of the length of the third side
   B. equal to the length of the third side
   C. less than the length of the third side
   D. greater than the length of the third side

   
   

3. The measure of an exterior angle of a triangle is always _________.
   A. equal to its adjacent interior angle
   B. less than its adjacent interior angle
   C. less than either remote interior angle
   D. greater than either remote interior angle

   
   

4. If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is ________________.
   A. equal to the third side of the second triangle.
   B. less than the third side of the second triangle.
   C. longer than the third side of the second triangle.
   D. greater than or equal to the third side of the second triangle.

   
   

5. Which of the following inequalities theorem is applicable in two triangles?
   A. Angle-Side Relationship
   B. SSS Inequality Theorem
   C. Triangle Inequality Theorem
   D. Exterior Angle Inequality Theorem

   
   

6. Which side of ∆ SET is the longest?
   a. $\overline{SE}$
   b. $\overline{ET}$
   c. $\overline{ST}$
   d. Cannot be determined

   
   

7. Which side is the shortest?
   a. $\overline{SE}$
   b. $\overline{ET}$
   c. $\overline{ST}$
   d. Cannot be determined

   
   

   [illustration: triangle SET with vertices E, T, and S; angle 93° at E and angle 40° at S]

   
   

8. What theorem is applied?
   a. Angle-Side Relationship
   b. SSS Inequality Theorem
   c. Triangle Inequality Theorem
   d. Exterior Angle Inequality Theorem

   
   

9. Which of the following angles is an exterior angle of ∆ EAR?
   a. ∠1
   b. ∠2
   c. ∠3
   d. ∠4

   
   

   [illustration: triangle EAR with vertices D, R, E, and A; angles numbered 1 to 6]

   
   

10. Which of the following could NOT be used as the length of the sides of a triangle?
    a. 4, 4, 4 c. 8, 14, 10
    b. 11, 12, 13 d. 6, 5, 12

    
    

11. List the sides of ∆TRI in order from longest to shortest measure.
    a. $\overline{FI}$, $\overline{FR}$, $\overline{RI}$
    b. $\overline{FI}$, $\overline{RI}$, $\overline{FR}$
    c. $\overline{FR}$, $\overline{RI}$, $\overline{FI}$
    d. $\overline{FR}$, $\overline{FI}$, $\overline{RI}$

    
    

[illustration: triangle TRI with vertices R, F, and I; angle 73° at R and angle 68° at F]

12. In the figure below, which of the following statements is true?
    a. ∠KIT = ∠ETI
    b. ∠KIT > ∠ITE
    c. ∠ETI > ∠KIT
    d. ∠IKT > ∠ITE

[illustration: triangle KIT with vertices I, K, and T; side KT extended beyond T to point E]

13. Hinges are used to fasten two things together and allow adjustment, rotation, twisting, or pivoting. Which of the following is NOT a hinged device?
    a. scissor c. screw
    b. folding ladder d. compass

    
    

14. Jason found out that two sides of two triangles are congruent. As he measures the included angle between these sides, he found out that the measure of the side opposite to the included angle of the first triangle is longer than the measure of the side to the included angle of the second triangle. What theorem did Jason use?
    a. Triangle Inequality Theorem
    b. Exterior Angle Inequality Theorem
    c. Hinge Theorem or SAS Inequality Theorem
    d. Converse of Hinge Theorem or SSS Inequality Theorem

    
    

15. Each of Angel, Anne, Luis and Sam was given a 15-inch piece of stick. They were instructed to create a triangle. Each cut the stick in their own chosen lengths as follows: Angel—5 in, 5 in, 5 in; Anne—3 in, 5 in, 7 in; Luis—6 in, 4 in, 5 in; and Sam—3 in, 4 in, 8 in. Who among them was not able to form a triangle?
    a. Angel c. Luis
    b. Anne d. Sam

    
    

Additional Activities

This section of this module will help you deepen your understanding about the importance of inequalities in triangle.

Activity 1

Directions: The distance Ronie walks from home to school is 130 meters and 100 meters when he goes to church from home. Romeo estimates that the distance Ronie walks when he goes directly to Church, coming from school, is 200 meters. Avelino’s estimation is 240 meters. Which estimation is feasible? Justify your answer. Write your answer on a separate sheet of paper.

[illustration: triangle with vertices School, Church, and Home; sides labeled 130 m (Home to School) and 100 m (Home to Church)]

To assess your answers, please refer to the rubric provided.

4	3	2	1
Use appropriate theorem with complete solution leading to correct answer.	Use appropriate theorem with solution but with incorrect answer.	Use inappropriate theorem with correct answer.	Use inappropriate theorem with incorrect answer.

Answer Key

[transcriber’s note: this page is printed rotated 180 degrees; the entries below are read from the rotated print and may not preserve the original layout]

What I Know

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

Assessment

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

What’s More

Activity 1

1. a. longest side – $\overline{AC}$
   b. shortest side – $\overline{AB}$
2. a. smallest – ∠Y; largest – ∠S
   b. Angle-Side Relationship Theorem
3. a. 153°
   b. > (both statement)
   c. Exterior Angle Inequality Theorem
4. a. >
   b. Converse of Hinge Theorem
5. a. <
   b. Hinge Theorem

Activity 2

1. X
2. ∆
3. ∆
4. ∆
5. X

[transcriber’s note: the marks in Activity 2 are printed as ∆ and X glyphs]

Activity 3

1. Greater
2. True
3. True
4. True
5. Opposite

What’s In

1. $\overline{EF}$, $\overline{BC}$
2. ∠C, ∠F
3. 180°, 180°
4. $\overline{EF} \cong \overline{BC}$
5. $\overline{DE} \cong \overline{AB}$

What I Have Learned

1. Angle-Side Relationship
2. Opposite
3. Triangle Inequality Theorem
4. Greater than
5. Exterior Angle Inequality
6. Remote interior angle
7. Hinge Theorem
8. longer than
9. SSS Inequality Theorem
10. included angle

What I Can Do

Activity 1

1. Dina, Angle-Side Relationship
2. No, Triangle Inequality Theorem
3. Alvin
4. Exterior Angle Inequality Theorem

Activity 2 Answers may vary.

What’s New

Measure Me: 25° ≈ T ∠, 90° F ≈ ∠, 65° ≈ I ∠ 30° ≈ M ∠, 100° A ∠, 50° ≈ X ∠ See Triangle Inequality Theorem 1 and 2

Compare Me:

1. 14 > 10
2. 18 > 6
3. 16 > 8
4. 13 > 10
5. 18 > 5
6. 15 > 8
   See Triangle Inequality Theorem 3

Complete Me: ∆ FLY: ∠1 ≈ 67°, ∠2 ≈ 132°, ∠3 ≈ 161°, ∠4 ≈ 48°, ∠5 ≈ 113°, ∠6 ≈ 19° ∆ FRE: ∠1 ≈ 70°, ∠2 ≈ 143°, ∠3 ≈ 147°, ∠4 ≈ 37°, ∠5 ≈ 110°, ∠6 ≈ 33° See Exterior Angle Inequality Theorem

Form Me: The length of the third side opposite 45° ≈ 5 inches long, while the length of the third side opposite 75° ≈ 8.5 inches long.

1. Congruent
2. Included angle
3. Triangle
4. See Hinge Theorem
   For 5 and 6, see Converse of Hinge Theorem

Note: Students’ answer may closely equal to the answer

Additional Activities

Applying Triangle Inequality Theorem 3, Avelino’s estimation is more feasible.

References

Abuzo, Emmanuel P., et.al, Mathematics- Grade 8 Learner’s Module First Edition, 2013. Published by the Department of Education

Abbie Brewer, “Triangle Inequality Task” Performance Based Learning and Assessment Task, accessed December 10, 2012, httpswww.radford.edurumath-smpdcPerformancesrcBrewer%20-%20Triangle%20Inequality%20Task.pdf

Gabriel, J.L., & Mallari, M.T.G. Mathematics for Grade 8 A Spiral Approach Explanations, Examples, Exercises, 2017. Educational Resources Corporation.

Geometry: Triangle Inequality and Angle-Side Relationship. (n.d.) Retrieved from https://www.onlinemathlearning.com/triangle-inequality.html

Kuta Software, “Infinite Geometry- Triangle Inequality Theorem, accessed 2015, httpswww.cpalms.orgPublicPreviewResourceLessonPreview40261

Triangle Inequality Theorem. (n.d.) Retrieved from https://www.mathsisfun.com/geometry/triangle-inequality-theorem

Usha Bhakuni, “Theorems of Inequality” Chapter 6 Lesson 6, accessed 2003-2020, https://study.com/academy/lesson/theorems-of-inequality.html#courseInfo

Usha Bhakuni, “Comparing Triangles with the Hinge Theorem” Chapter 6 Lesson 8, accessed 2003-2020, https://study.com/academy/lesson/comparing-triangles-with-the-hinge-theorem.html

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