Mathematics 8 Quarter 4 Module 2: Applying Theorems on Triangle Inequalities

Mathematics Quarter 4 – Module 2 Applying Theorems on Triangle Inequalities

[illustration: teacher and students in a classroom and outdoors under a tree, with a globe and a lamp]

CO_Q4_Mathematics 8_ Module 2

Mathematics – Grade 8 Alternative Delivery Mode Quarter 4 – Module 2 Applying Theorems on Triangle Inequalities First Edition, 2020

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Mathematics Quarter 4 – Module 2 Applying Theorems on Triangle Inequalities

8

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

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This module was designed and written with you in mind. It is here to help you master the skills of applying theorems on triangle inequalities. 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 applying theorems on triangle inequalities (M8GE-IVb-1).

After going through this module, you are expected to:

1. demonstrate understanding about the theorems on triangle inequalities;
2. apply theorems on triangle inequalities to find measures of angles and sides of a triangle; and
3. manifest appreciation in applying theorems on triangle inequalities to real-life situations.

What I Know

[illustration: hand holding a pencil]

Directions: Read and answer each item carefully. Choose the letter of the correct answer. Write your answers on a separate sheet of paper.

1. Carlos concluded that the longest side in 𝛥𝑀𝐴𝑃 is 𝐴𝑃̅̅̅̅ after finding that the angle opposite to 𝐴𝑃̅̅̅̅ is the largest angle. What theorem did Carlos use to make such conclusion?

   • A. Triangle Inequality Theorem 1 (𝑆𝑠 → 𝐴𝑎)
   • B. Triangle Inequality Theorem 2 (𝐴𝑎 → 𝑆𝑠)
   • C. Hinge Theorem or SAS Triangle Inequality Theorem
   • D. Converse of Hinge Theorem or SSS Triangle Inequality Theorem

2. Liza, Kathryn, and Nadine were each given a 21-inch piece of stick. They were instructed to create a triangle. Each stick was cut in their own chosen lengths as follows: Liza – 6 𝑖𝑛, 7 𝑖𝑛, 8 𝑖𝑛; Kathryn – 4 𝑖𝑛, 6 𝑖𝑛, 11 𝑖𝑛; and Nadine – 3 𝑖𝑛, 5 𝑖𝑛, 13 𝑖𝑛. Who among them was able to make a triangle?

   • A. Liza
   • B. Nadine
   • C. Kathryn
   • D. All of them

For items 3 – 4, refer to ∆𝑷𝑬𝑵 below.

[chart: triangle; ∆𝑷𝑬𝑵 (unnumbered). Vertices: E (top), P (bottom left), N (bottom right). Side labels: PE = 11 cm; EN = 14 cm; PN = 18 cm.]

3. What is the largest angle?

   • A. ∠𝐸
   • B. ∠𝑁
   • C. ∠𝑃
   • D. Cannot be determined.

4. What is the smallest angle?

   • A. ∠𝐸
   • B. ∠𝑁
   • C. ∠𝑃
   • D. Cannot be determined.

For items 5 – 7, consider the triangle ∆TIP at the right.

[chart: triangle; ∆TIP (unnumbered). Vertices: I (top), T (bottom left), P (bottom right). Angle labels: at I = 104°; at T = 33°.]

5. Which side of ∆𝑇𝐼𝑃 is the longest?

   • A. 𝑇𝐼̅̅̅
   • B. 𝐼𝑃̅̅̅
   • C. 𝑇𝑃̅̅̅̅
   • D. Cannot be determined.

6. Which side of ∆𝑇𝐼𝑃 is the shortest?

   • A. 𝑇𝐼̅̅̅
   • B. 𝐼𝑃̅̅̅
   • C. 𝑇𝑃̅̅̅̅
   • D. Cannot be determined.

7. What theorem did you use in determining the longest and shortest side of ∆𝑇𝐼𝑃?

   • A. Triangle Inequality Theorem 1 (𝑆𝑠 → 𝐴𝑎)
   • B. Triangle Inequality Theorem 2 (𝐴𝑎 → 𝑆𝑠)
   • C. Hinge Theorem or SAS Triangle Inequality Theorem
   • D. Converse of Hinge Theorem or SSS Triangle Inequality Theorem

8. Which of the following lengths in cm are possible measures of the sides of a triangle?

   • A. 1, 2, 3
   • B. 4, 5, 10
   • C. 5, 6, 7
   • D. 5, 9, 15

9. Which of the following theorems will support your answer in number 8?

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

For items 10 – 12, consider the figure at the right.

[illustration: figure with points K, N, I and P; K lies on a line extended beyond N, N is joined to P and to I, and P is joined to I]

10. Which of the following statements is true?

    • A. 𝑚∠𝐾𝑁𝐼 = 𝑚∠𝑃𝐼𝑁.
    • B. 𝑚∠𝐾𝑁𝐼 > 𝑚∠𝑃𝐼𝑁.
    • C. 𝑚∠𝐾𝑁𝐼 < 𝑚∠𝑃𝐼𝑁.
    • D. Cannot be determined.

11. Which of the following statements is true?

    • A. 𝑚∠𝑁𝑃𝐼 = 𝑚∠𝐼𝑁𝐾.
    • B. 𝑚∠𝑁𝑃𝐼 > 𝑚∠𝐼𝑁𝐾.
    • C. 𝑚∠𝑁𝑃𝐼 < 𝑚∠𝐼𝑁𝐾.
    • D. Cannot be determined.

12. What theorem did you apply to answer item numbers 10 and 11?

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

13. The measures of the angles of ∆𝐷𝐸𝐹 are as follows: 𝑚∠𝐷 = 2𝑥 + 3⁰; 𝑚∠𝐸 = 2𝑥 + 5⁰; and 𝑚∠𝐹 = 𝑥 – 8⁰. Arrange the sides in increasing order of length.

    • A. 𝐷𝐸̅̅̅̅, 𝐸𝐹̅̅̅̅, 𝐷𝐹̅̅̅̅
    • B. 𝐸𝐹̅̅̅̅, 𝐷𝐸̅̅̅̅, 𝐷𝐹̅̅̅̅
    • C. 𝐷𝐸̅̅̅̅, 𝐷𝐹̅̅̅̅, 𝐸𝐹̅̅̅̅
    • D. 𝐷𝐹̅̅̅̅, 𝐷𝐸̅̅̅̅, 𝐸𝐹̅̅̅̅

For items 14 – 15, consider the given ∆PET and ∆DOG below.

[chart: two triangles; ∆PET and ∆DOG (unnumbered). ∆PET: vertex E at top with angle 32°, base PT. ∆DOG: vertex O at top with angle 48°, base DG. Tick marks show PE and ET congruent-marked segments, and OD and OG marked with matching tick marks.]

14. What can you conclude in the given figures?

    • A. |𝑃𝑇| = |𝐷𝐺|
    • B. |𝑃𝑇| > |𝐷𝐺|
    • C. |𝐷𝐺| < |𝑃𝑇|
    • D. |𝐷𝐺| > |𝑃𝑇|

15. What theorem did you use to answer item number 14?

    • A. Exterior Angle Inequality Theorem
    • B. Triangle Inequality Theorem 1 (𝑆𝑠 → 𝐴𝑎)
    • C. Hinge Theorem or SAS Triangle Inequality Theorem
    • D. Converse of Hinge Theorem or SSS Triangle Inequality Theorem

Lesson 1: Applying Theorems on Triangle Inequalities

Have you ever thought of how artists use triangles in their artworks? Have you ever wondered how builders, architects, and engineers use triangular features in their designs? This module reveals the answers to these questions.

[illustration: radio tower with a small building at its base]

[illustration: two children on a seesaw]

[illustration: billiard balls arranged in a triangle with a cue ball]

[illustration: excavator]

[illustration: pair of scissors]

[illustration: roof truss with triangular bracing]

[transcriber’s note: this excerpt continues a module whose earlier pages were transcribed separately. It begins at the printed page 5 (“What’s In”), so the lesson-level heading is not repeated here.]

What’s In

Activity: Find Our Connection

Directions: Connect the if-clause in Column A with the then-clause in Column B to create a statement describing the theorem of triangle inequalities listed in each item below. The first number is done for you. Write your answer on a separate sheet of paper.

[chart: matching diagram; Activity: Find Our Connection. The three columns are not aligned row by row, and the connecting lines are not transcribed. Each column is listed separately below.]

THEOREM

1. Hinge Theorem or SAS Inequality Theorem
2. Triangle Inequality Theorem 1 (Ss→Aa)
3. Exterior Angle Theorem
4. Triangle Inequality Theorem 2 (Aa→Ss)
5. Converse of Hinge Theorem or SSS Inequality Theorem
6. Triangle Inequality Theorem 3 (S₁ +S₂ >S₃)

COLUMN A

• If one side of a triangle is longer than a second side,
• 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,
• If one angle of a triangle is larger than a second angle,
• If the angle is exterior,
• If the lengths of two sides of a triangle are added,
• 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,

COLUMN B

• then the angle opposite the longer side is larger than the angle opposite the shorter side.
• then the third side of the first triangle is longer than the third side of the second.
• then the side opposite the larger angle is longer than the side opposite the smaller angle.
• then the included angle of the first triangle is larger than the included angle of the second.
• then its measure is greater than the measure of either remote interior angle of a triangle.
• then the sum of their measures is greater than the measure of the third side.

What’s New

Activity: Follow My Steps

Materials: broom stick, ruler, pen/pencil, cutter

Directions: Do as directed, then answer the questions that follow. Write your answer on a separate sheet of paper.

1. Prepare a stick measuring 12 inches. (Note: The figure is not drawn to scale.)

[illustration: a broom stick above a ruler marked 0 to 12]

2. Using a cutter, cut the stick into parts measuring 3 inches, 4 inches, and 5 inches respectively.

[illustration: three sticks above rulers marked 0 to 3, 0 to 4, and 0 to 5]

3. Position the sticks as shown in the illustration. Simultaneously move the two shorter lengths up then down.

[illustration: a horizontal stick with two shorter sticks rising from its ends, with arrows showing them moving up and then down]

1. 3 inches, 4 inches,
2. 3 inches, 4 inches,

[transcriber’s note: the two lines above are printed in the figure area of the page, below the illustration, and may be stray labels from the figure.]

Questions

1. What figure were you able to form?
2. Was there a chance that the shorter lengths (3 inches and 4 inches) met as you moved them down
3. Can you still form the same figure if the sticks measure 2 inches, 3 inches, and 7 inches, respectively? Explain your answer.

What is It

Theorems on triangle inequalities are categorized into two. These are the inequalities in one triangle and inequalities in two triangles. For inequalities in one triangle, the following theorems apply:

1. Triangle Inequality Theorem 1 (𝑆𝑠 → 𝐴𝑎)
2. Triangle Inequality Theorem 2 (𝐴𝑎 → 𝑆𝑠)
3. Triangle Inequality Theorem 3 (𝑆₁ + 𝑆₂ > 𝑆₃)
4. Exterior Angle Inequality Theorem.

On the other hand, inequalities in two triangles use either of these theorems:

1. Hinge Theorem or SAS Inequality Theorem
2. Converse of Hinge Theorem or SSS Inequality Theorem.

Triangle Inequality Theorem 1 (Ss→Aa)

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 shorter side.

Example 1

List down the angles of ∆𝐺𝑂𝑇 from greatest to least measure. (Note: The figure is not drawn to scale.)

[chart: triangle diagram; Example 1 figure ∆GOT. Vertices: O (top), G (bottom left), T (bottom right). Side labels: OG = 6, OT = 7, GT = 8.]

Solution

In the figure, 𝐺𝑇̅̅̅̅ is the longest side, thus, the angle opposite it which is ∠𝑂, has the greatest measure. Since 𝑂𝐺̅̅̅̅ is the shortest side, hence the angle opposite it which is ∠𝑇, has the least measure.

Listing the angles of ∆𝐺𝑂𝑇 from greatest to least measure results to

∠𝑂, ∠𝐺, ∠𝑇.

[transcriber’s note: a stray “[” character appears above Example 1 on this page and is omitted.]

Triangle Inequality Theorem 2 (Aa→Ss)

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 smaller angle.

Example 1

List down the sides of ∆𝐿𝑂𝐷 from longest to shortest. (Note: The figure is not drawn to scale.)

[chart: triangle diagram; Example 1 figure ∆LOD. Vertices: O (top), L (bottom left), D (bottom right). Angle labels: at O = 110°, at L = 40°, at D = 30°.]

Solution

Writing the angles from the largest to the smallest measure gives ∠𝑂, ∠𝐿, ∠𝐷. Applying Triangle Inequality Theorem 2, it can be concluded that the longest side is 𝐿𝐷̅̅̅̅, 𝑂̅̅̅𝐷̅ is the second longest, and ̅𝐿𝑂̅̅̅ is the shortest side.

Example 2

List down the sides of ∆𝐵𝐶𝐴 from the shortest to the longest measure. (Note: The figure is not drawn to scale.)

[chart: triangle diagram; Example 2 figure ∆BCA. Vertices: B (top), C (bottom left), A (bottom right). Angle labels: at C = 43°, at A = 63°.]

Solution

The first thing to do is to find the unknown measure of ∠𝐵. Note that the sum of the measures of the interior angles of a triangle is 180⁰. To solve for the measure of ∠𝐵, we use the equation 𝑚∠𝐵 + 𝑚∠𝐶 + 𝑚∠𝐴 = 180°.

Equation	Reason
𝑚∠𝐵 + 43° + 63° = 180°	Use Substitution
𝑚∠𝐵 + 106° = 180°	Simplify
𝑚∠𝐵 + 106° − 106° = 180° − 106°	Use Subtraction Property of Equality
𝑚∠𝐵 + 0 = 180° − 106°	Use Identity Property for Addition
𝑚 ∠𝐵 = 74°	Simplify

Since ∠𝐵 has the greatest measure of 74°, it can be concluded that the side opposite it which is 𝐴𝐶̅̅̅̅ is the longest, followed by 𝐵𝐶̅̅̅ since ∠𝐴 is the second largest, and 𝐴𝐵̅̅̅̅ is the shortest side since ∠𝐶 has the least measure. Listing down the sides of ∆𝐵𝐶𝐴 from shortest to longest measure gives 𝐴𝐵̅̅̅̅, 𝐵𝐶̅̅̅, 𝐴𝐶̅̅̅̅.

Triangle Inequality Theorem 3 (S₁ +S₂ >S₃)

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

Illustration

Consider ∆𝐸𝐹𝐺 as shown below, with 𝑒, 𝑓, and 𝑔 as the side lengths.

[chart: triangle diagram; Illustration figure ∆EFG. Vertices: F (top), E (bottom left), G (bottom right). Side labels: EF = g, FG = e, EG = f.]

The triangle inequality theorem 3 states that: i) 𝑓 + 𝑔 > 𝑒 ii) 𝑒 + 𝑔 > 𝑓 iii) 𝑒 + 𝑓 > 𝑔

To be able to form a triangle, each of the three inequalities must be true. If one of these inequalities is false, then it is not possible for these lengths to form a triangle. Consider the following examples.

Example 1

Check whether it is possible to form a triangle with lengths 13 cm, 14 cm, and 22 cm.

Solution:

For 13 cm, 14 cm and 22 cm to be considered as side lengths of a triangle, these should satisfy the inequality theorem (S₁ +S₂ >S₃).

Hence,

Inequality	Simplified	Result
13 + 14 > 22	27 > 22	True
13 + 22 > 14	35 > 14	True
14 + 22 > 13	36 > 13	True

All the three conditions are satisfied, therefore it is possible to form a triangle with lenghts 13 cm, 14 cm, and 22 cm.

Example 2

Tell whether it is possible to form a triangle with lengths 2 cm, 8 cm, and 12 cm.

Solution

For 2 cm, 8 cm and 12 cm to be considered as side lengths of a triangle, these should satisfy the inequality theorem (S₁ +S₂ >S₃).

Thus,

Inequality	Simplified	Result
2 + 8 > 12	10 > 12	False
2 + 12 > 8	14 > 8	True
8 + 12 > 2	20 > 2	True

One of the inequalities is false. Therefore, it is not possible to form a triangle with lengths 2 cm, 8 cm, and 12 cm.

Example 3

If two of the sides of a triangular frame measure 8 inches and 15 inches, find the possible lengths of the third side 𝑥.

Solution

To find the possible lengths of the third side of the triangular frame, you can use the formula:

𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ𝑠 𝑜𝑓 𝑡𝑤𝑜 𝑠𝑖𝑑𝑒𝑠 < 𝑈𝑛𝑘𝑛𝑜𝑤𝑛 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑠𝑖𝑑𝑒 < 𝑆𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑠𝑖𝑑𝑒𝑠.

In the situation, the length of the third side is represented by the variable 𝑥.

𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑠𝑖𝑑𝑒𝑠 < 𝑥 < 𝑆𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑠𝑖𝑑𝑒𝑠

Equation	Reason
15 − 8 < 𝑥 < 15 + 8	Use Substitution
7 < 𝑥 < 23	Simplify

The third side can have length between 7 inches and 23 inches.

To check whether the length of the third side 𝑥 could be any value between 7 inches and 23 inches, choose one value in the interval and apply the Triangle Inequality Theorem 3 (S₁ +S₂ >S₃). Consider, 𝑥 = 10 inches.

Inequality	Simplified	Result
8 + 15 > 10	23 > 10	True
8 + 10 > 15	18 > 15	True
15 + 10 > 8	25 > 8	True

Therefore, the length of the third side can be any value between 7 inches and 23 inches.

Exterior Angle Inequality Theorem

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

Remember

Exterior Angle – an angle that forms a linear pair with one of the interior angles of a triangle.

Remote Interior Angle – an angle of a triangle that is not adjacent to a specified exterior angle.

Linear Pair Theorem – if two angles form a linear pair, then the two angles are supplementary and adjacent.

Example 1

Given the figure below, name the following: a. linear pair b. exterior angle c. remote interior angles of ∠𝐴𝐶𝐷

[chart: triangle diagram; Example 1 figure. Vertices: A (top), B (bottom left), C (bottom middle), with D on the extension of BC to the right.]

Solution:

a. ∠𝐴𝐶𝐷 and ∠𝐴𝐶𝐵 form a linear pair. b. ∠𝐴𝐶𝐷 is an exterior angle. c. ∠𝐴 𝑎𝑛𝑑 ∠𝐵 are the remote interior angles of ∠𝐴𝐶𝐷.

Example 2

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

a. Find the measure of ∠𝑆𝑀𝑅. b. Compare the measure of ∠𝑆𝑀𝑅 to the measure of each remote interior angle of ∆𝐴𝑀𝑅.

[chart: triangle diagram; Example 2 figure. Vertices: M (upper middle), A (bottom left), R (bottom right). S lies on the extension of AM beyond M. Angle labels: at M (between A and R) = 75°, at R = 45°.]

Solution:

a. In the figure, notice that ∠𝑆𝑀𝑅 and ∠𝐴𝑀𝑅 form a linear pair. This means that the sum of the measures of these angles is 180°. To find the 𝑚∠𝑆𝑀𝑅,

Equation	Reason
𝑚∠𝑆𝑀𝑅 + 𝑚∠𝐴𝑀𝑅 = 180°	Apply Linear Pair Theorem
𝑚∠𝑆𝑀𝑅 + 75° = 180°	Use Substitution
𝑚∠𝑆𝑀𝑅 + 75° − 75° = 180° − 75°	Use Subtraction Property of Equality
𝑚∠𝑆𝑀𝑅 + 0 = 180° − 75°	Use Identity Property for Addition
𝑚∠𝑆𝑀𝑅 = 105°	Simplify

Hence, 𝑚∠𝑆𝑀𝑅 = 105°.

b. Find the measure of ∠𝐴 of ∆𝐴𝑀𝑅.

Equation	Reason
𝑚∠𝐴 + 𝑚∠𝑀 + 𝑚∠𝑅 = 180°
𝑚∠𝐴 + 75° + 45° = 180°	Use Substitution
𝑚∠𝐴 + 120° = 180°	Simplify
𝑚∠𝐴 + 120° − 120° = 180° − 120°	Use Subtraction Property of Equality
𝑚∠𝐴 + 0 = 180° − 120°	Use Identity Property for Addition
𝑚∠𝐴 = 60°	Simplify

In the figure, ∠𝑆𝑀𝑅 is an exterior angle which has a measure that is greater than either measure of its remote interior angle, ∠𝐴 or ∠𝑅. In symbols,

Inequality	Check	Result
𝑚∠𝑆𝑀𝑅 > 𝑚∠𝐴	105° > 60°	True
𝑚∠𝑆𝑀𝑅 > 𝑚∠𝑅	105° > 45°	True

Further, adding the measures of these two remote interior angles will give the measure of the exterior angle. In symbols,

𝑚∠𝐴 + 𝑚∠𝑅 = 𝑚∠𝑆𝑀𝑅 60° + 45° = 105°

Example 3

Situation: Four billiard balls are left on the table as shown below. Use the expressions to determine the measure of the exterior angle. (Note: The figure is not drawn to scale.)

[transcriber’s note: the figure for Example 3 is not on this page image. The word “and” printed between the two inequality checks above is kept as printed.]

Solution:

(2𝑥 + 25⁰) + (𝑥 − 15⁰) + (𝑥 + 10⁰) = 180⁰ Apply Sum of the Interior Angles of a Triangle is 180°

(2𝑥 + 𝑥 + 𝑥) + (25⁰ − 15⁰ + 10⁰) = 180⁰ Combine like terms

4𝑥 + 20 = 180° Simplify

4𝑥 + 20⁰ − 20⁰ = 180° − 20⁰ Use Subtraction Property of Equality

4𝑥 + 0 = 160° Use Identity Property for Addition

4𝑥 = 160° Simplify

(¼) 4𝑥 = (¼) 160° Use Multiplication Property of Equality

𝑥 = 40° Simplify

Now, substitute 𝑥 by its value in the given expressions.

𝑚∠𝐴 = 2𝑥 + 25⁰ = 2(40⁰) + 25⁰ = 80⁰ + 25⁰ = 105⁰

𝑚∠𝐵 = 𝑥 − 15⁰ = 40⁰ − 15⁰ = 25⁰

𝑚∠𝐶 = 𝑥 + 10 = 40⁰ + 10⁰ = 50⁰

The measures of ∠𝐴, ∠𝐵 and ∠𝐶 are 105°, 25° and 50°, respectively.

To find the measure of the exterior angle, you can either do the following solutions:

a. Sum of the measures of the remote interior angles 25° + 105° = 130°

b. Linear Pair Theorem 180° − 50° = 130°

[illustration: billiard balls on a table, with three balls at the vertices of a triangle labeled A, B, C, and a fourth ball labeled D along a line]

𝑚∠𝐴 = 2𝑥 + 25⁰ 𝑚∠𝐵 = 𝑥 − 15⁰ 𝑚∠𝐶 = 𝑥 + 10⁰

Hinge Theorem or SAS Inequality

If two sides of one triangle are congruent to two sides of a second triangle, and 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.

Example 1

Consider ∆𝐴𝐵𝐶 and ∆𝑋𝑌𝑍. Describe the lengths of sides 𝐴𝐶̅̅̅̅ and 𝑌𝑍̅̅̅̅. (Note: The figure is not drawn to scale.)

[illustration: two triangles side by side; triangle ABC with vertex A at top, B at lower left with angle 45°, C at lower right; triangle XYZ with vertex Y at top, X at lower left with angle 60°, Z at lower right; tick marks show congruent sides AB and YX, and BC and XZ]

Solution

As shown in the figure, sides 𝐴𝐵̅̅̅̅ and 𝐵𝐶̅̅̅ of ∆𝐴𝐵𝐶 are congruent respectively to sides 𝑌𝑋̅̅̅̅ and 𝑋𝑍̅̅̅̅ of ∆𝑋𝑌𝑍. Applying the Hinge Theorem, 𝐴𝐶̅̅̅̅ is shorter than 𝑌𝑍̅̅̅̅ since the opposite angle to 𝐴𝐶̅̅̅̅ which is ∠𝐵 has a smaller measure than ∠𝑋 which is opposite to 𝑌𝑍̅̅̅̅.

Example 2

Situation: The figures below show two pairs of scissors of the same size in two different positions. In which figure is the distance between the tips of the two blades greater? Use the Hinge Theorem to justify your answer. (Note: The figure is not drawn to scale.)

[illustration: Figure 1, a pair of scissors with the blades open at 50°] [illustration: Figure 2, a pair of scissors with the blades open at 26°]

Figure 1 Figure 2

Solution:

The distance between the tips of the two blades in Figure 1 is greater since the angle opposite this side has a greater measure which is 50° as compared to that in Figure 2 which is only 26°.

Converse of Hinge Theorem or SSS Inequality Theorem

If two sides of one triangle are congruent to two sides of a second triangle, and 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.

Example 1

Consider ∆𝑀𝑂𝑃 and ∆𝑄𝑅𝑆. Which angle is larger, ∠𝑀 or ∠𝑆?

[chart: two triangles; triangle MOP with vertices O (top), M (lower left), P (lower right), side OP labeled 16, tick marks on sides MO and MP; triangle QRS with vertices Q (top), R (lower left), S (lower right), side RQ labeled 9, tick marks on sides QS and RS; the figure is not drawn to scale]

Solution:

In the figure |𝑀𝑂| = |𝑆𝑄|, |𝑀𝑃| = |𝑆𝑅| and |𝑂𝑃| > |𝑄𝑅| or 16 > 9. Thus, by the use of SSS Inequality Theorem, the measure of ∠𝑀 is greater than the measure of ∠𝑆.

Example 2

Situation: Apply the SSS Inequality Theorem to determine which figure has a greater measure of the included angle. (Note: The figure is not drawn to scale.)

[illustration: Figure 1, an excavator with its arm forming a triangle, the distance between the two joints labeled 8 m] [illustration: Figure 2, an excavator with its arm forming a triangle, the distance between the two joints labeled 5 m]

Figure 1 Figure 2

Solution:

Applying the Converse of Hinge Theorem or SSS Inequality Theorem makes the angle opposite the 8-meter side length larger than the angle opposite the 5-meter side length of a triangle. Thus, the included angle in Figure 1 has a greater measure than the included angle in Figure 2.

What’s More

[illustration: hand placing blocks in a stack]

Let us deepen your understanding on triangle inequality theorems by performing the following activities. Write your answers on a separate sheet of paper. (Note: All figures are not drawn to scale.)

Activity 1

Directions: Arrange the angles of ∆𝐴𝐵𝐶 from greatest to least measure given the lengths of its sides.

1. |𝐴𝐵| = 5 𝑐𝑚, |𝐵𝐶| = 10 𝑐𝑚, |𝐴𝐶| = 12 𝑐𝑚
2. |𝐵𝐴| = 10 𝑐𝑚, |𝐶𝐵| = 19 𝑐𝑚, |𝐶𝐴| = 11 𝑐𝑚
3. |𝐴𝐶| = 7 𝑐𝑚, |𝐴𝐵| = 3 𝑐𝑚, |𝐵𝐶| = 5 𝑐𝑚

Activity 2

A. Use the Triangle Inequality Theorem 3 (𝑆1 + 𝑆2 > 𝑆3) in determining whether the following numbers in cm can be side lengths of a triangle. Put a check mark (√) to indicate your answer.

Numbers (cm)	Yes	No
1) 9, 13, 20
2) 3, 4, 8
3) 4, 25, 30

B. Find the possible values for the length of the third side 𝑥 using the Triangle Inequality Theorem 3 (𝑆1 + 𝑆2 > 𝑆3).

1. 14, 36 2) 8, 21 3) 13, 40

Activity 3: tri-G L E (Greater than, Less than, or Equal)

Directions: Study the figure below and use >, <, or = to compare the measures of angles and sides.

[illustration: triangle with vertices R (top), A (right), and C (lower left); angle at R labeled 40°, angle at C labeled 75°; point E lies on the extension of line RC below C]

1. 𝑚∠𝑅 ______ 𝑚∠𝐴
2. 𝑚∠𝑅𝐶𝐴 ______ 𝑚∠𝑅𝐴𝐶
3. 𝑚∠𝐸𝐶𝐴 ______ 𝑚∠𝐶𝑅𝐴
4. 𝑚∠𝐶𝐴𝑅 ______ 𝑚∠𝐸𝐶𝐴
5. 𝑚∠𝑅 + 𝑚∠𝐴 ______ 𝑚∠𝐴𝐶𝐸
6. |𝑅𝐴| ______ |𝐶𝑅|
7. |𝐴𝐶| ______ |𝐶𝑅|

Activity 4:

Directions: Consider the situation below and answer the questions that follow.

Situation: Anna and Leo start from the same point and decided to walk in opposite directions. Anna walks 50 meters East then took another 30 meters South. Leo walks 30 meters West. He then takes a right turn of 40° and walks another 50 meters as shown in the figure below.

[illustration: Anna and Leo at the two ends of a path; Anna’s path is 50 m, then 30 m; Leo’s path is 30 m west, then a 40° turn and 50 m, ending at a right angle with 30 m; a dashed horizontal line marks the starting direction]

Questions

1. Who is farther from the point where they started? Support your answer.
2. What theorem/s is/are applied in the situation?

Activity 5:

Directions: Study the illustration and answer the questions that follow.

[illustration: a turtle and a dog on a horizontal line labeled West (left) and East (right); a 5-meter path from the turtle goes up to the midpoint; an 8-meter path from the midpoint goes up to the dog; the angle at the turtle is labeled 105° and the angle between the line and the dog’s path is labeled 𝑦; the segments marked with equal tick marks are congruent; the angle at the turtle is labeled 𝑥]

Questions

1. In the illustration, which angle between 𝑥 and 𝑦 has a greater measure? Support your answer.
2. What theorem/s is/are applied in the situation?

What I Have Learned

[illustration: hand holding a light bulb]

Let us synthesize the things that you’ve learned in this module by performing the next activity. Write your answers on a separate sheet of paper. (Note: All figures are not drawn to scale.)

Directions: Give what is asked in each of the following figures.

A. [chart: triangle; vertices A (lower left), B (top), C (right); side AB labeled 10, side BC labeled 12, side AC labeled 13]

1. Which angle has the greatest measure?
2. What theorem supports your answer?
3. Arrange the angles of 𝛥𝐴𝐵𝐶 from least to greatest measure.

B. [chart: triangle XYZ; vertex Y (top), X (lower left) with angle 55°, Z (lower right) with angle 35°]

1. What is the measure of ∠𝑌?
2. Which segment is shortest? longest?
3. What theorem justifies your answer in question 2?

C. [chart: triangle with sides labeled 5 and 11 and third side labeled 𝑥]

1. What is the range of possible measures of the third side 𝑥?
2. What theorem is applied in the problem?

D. [chart: triangle ABC with vertex B at top having angle 85°, vertex A at lower right having angle 30°, and vertex C on a horizontal line; the line extends from D through C to A; the exterior angle is at C]

1. Name the exterior angle.
2. What is the measure of ∠𝐵𝐶𝐷?
3. Compare the measure of the exterior angle a) to the sum of the measures of the remote interior angles, b) to the measure of either remote interior angle.

E. [chart: triangle POR with vertex P at top, O at lower left, R at lower right; point Q lies on side OR; angle at Q labeled 85°; tick marks show OQ and QR are congruent]

1. What is the measure of ∠𝑅𝑄𝑃?
2. Between 𝑂𝑃̅̅̅̅ and 𝑃𝑅̅̅̅̅, which is longer?
3. What theorem justifies your answer?

F. [chart: two triangles; triangle CTA with sides CT and CA marked congruent, and side AT labeled 30; triangle GOD with sides GO and GD marked congruent, and side DO labeled 32; tick marks indicate the congruent sides]

1. Between ∠𝐶 and ∠𝐺, which is larger?
2. What theorem supports your answer?

What I Can Do

[illustration: hands assembling hexagonal bolts]

Directions: Read and analyze the given situation and answer the following questions. Write your answers on a separate sheet of paper.

Activity 1: Intersection Ahead!

Situation: Three roads intersect at certain points as illustrated in the figure below.

[chart: triangular road layout; roads form triangle with vertices M (lower left), A (top), and P (lower right), with an extension beyond P to R on the horizontal road and a point S on the extension of the left road beyond A; angle at M labeled 75°, angle at P labeled 50°; a STOP sign, a tree, and a seesaw are drawn along the roads]

Questions

1. What is the measure of ∠𝐴?
2. List down the angles of 𝛥𝑀𝐴𝑃 from least to greatest measure.
3. Arrange the sides of 𝛥𝑀𝐴𝑃 in ascending order.
4. Between 𝑀 and 𝐴, which intersection is closer to 𝑃?
5. List down the exterior angles found in the illustration.
6. Which of these exterior angles has a measure of 125°?

Activity 2: Time’s Up!

One of the real-life applications of Hinge Theorem is seen in clocks. In the illustration below, each clock displays different time.

Directions: Observe each clock and answer the questions that follow.

Questions

1. Do the lengths of the hands of the clock change as the time (hour) changes?
2. What do you observe about the measures of the angles formed by the hands of the clock at different hours?
3. What affects the measure of the distance between the tips of the hands of the clock? Explain.

[illustration: three analog clocks showing different times, with hour and minute hands]

Assessment

Directions: Read and answer each of the questions carefully. Write the letter that corresponds to the correct answer on a separate sheet of paper. (Note: The figures are not drawn to scale.)

1. Which is always true about the measure of an exterior angle of a triangle?
   a. It is less than the measure of its adjacent interior angle.
   b. It is less than the measure of either remote interior angle.
   c. It is greater than the measure of its adjacent interior angle.
   d. It is greater than the measure of either remote interior angle.
2. Which of the following numbers of the same unit could represent the side lengths of a triangle?
   a. 1, 3, 5 c. 3, 6, 10
   b. 2, 4, 5 d. 4, 5, 11

For items 3 – 5, refer to the ∆POD at the right.

3. Which is the largest angle?
   a. ∠D
   b. ∠O
   c. ∠P
   d. Cannot be determined

[chart: triangle diagram; ∆POD with vertices O (top), P (bottom left), D (bottom right); side labels 21, 19, 30]

4. Which is the smallest angle?
   a. ∠D c. ∠P
   b. ∠O d. Cannot be determined.
5. What theorem is applicable in determining the smallest and largest angles of ∆POD?
   a. Triangle Inequality Theorem 1 (Ss→Aa)
   b. Triangle Inequality Theorem 2 (Aa→Ss)
   c. Hinge Theorem or SAS Triangle Inequality Theorem
   d. Converse of Hinge Theorem or SSS Triangle Inequality Theorem

For items 6 – 8, refer to ∆SAW at the right.

6. Which side of ∆SAW is the shortest?
   a. $\overline{AS}$
   b. $\overline{SW}$
   c. $\overline{WA}$
   d. Cannot be determined

   
   

7. Which side of ∆SAW is the longest?
   a. $\overline{AS}$ b. $\overline{SW}$ c. $\overline{WA}$ d. Cannot be determined.

   
   

8. What theorem is applicable in determining the shortest and longest sides of ∆SAW?
   a. Triangle Inequality Theorem 1 (Ss→Aa)
   b. Triangle Inequality Theorem 2 (Aa→Ss)
   c. Hinge Theorem or SAS Triangle Inequality Theorem
   d. Converse of Hinge Theorem or SSS Triangle Inequality Theorem

   
   

9. Which of the following could be a possible measure of the third side of the triangle if the two sides measure 12 meters and 25 meters?
   a. 11 meters b. 13 meters c. 28 meters d. 37 meters

   
   

10. Given below is ∆ART. List the sides from the longest to the shortest.
    a. $\overline{AT}, \overline{AR}, \overline{RT}$
    b. $\overline{AR}, \overline{TR}, \overline{TA}$
    c. $\overline{RT}, \overline{RA}, \overline{TA}$
    d. $\overline{TA}, \overline{TR}, \overline{AR}$

    
    

[chart: triangle diagram; ∆SAW with vertices S (top), A (bottom left), W (bottom right); angle labels 48° (at A), 85° (at W)]

[chart: triangle diagram; ∆ART with vertices R (top), A (bottom left), T (bottom right); angle labels 70° (at R), 52° (at A), 58° (at T)]

For items 11 – 13, consider the figure below.

[chart: diagram; triangle RAD with ray RG extended from R, ray DE extended from D; angle labels 105° (at R) and 155° (at D)]

11. What is the measure of ∠A?
    a. 25° b. 50° c. 105° d. 155°
12. What is the measure of ∠GRD?
    a. 25° b. 75° c. 105° d. 155°
13. Which is the longest side of ∆RAD?
    a. $\overline{AR}$ b. $\overline{DA}$ c. $\overline{RD}$ d. Cannot be determined.
14. What theorem supports this statement, “In an obtuse triangle, the longest side is opposite the obtuse angle.”
    a. Triangle Inequality Theorem 1 (Ss → Aa)
    b. Triangle Inequality Theorem 2 (Aa → Ss)
    c. Hinge Theorem or SAS Triangle Inequality Theorem
    d. Converse of Hinge Theorem or SSS Triangle Inequality Theorem
15. You are asked to fence a triangular lot. Two sides of the lot have lengths 20 meters and 14 meters. What is the maximum whole number of meters of fence do you possibly need?
    a. 34 meters b. 35 meters c. 67 meters d. 68 meters

Additional Activities

Directions: Answer the following activities as indicated. Write your answers on a separate sheet of paper.

Activity 1

In ∆DEF, m∠D = 45° and the exterior angle ∠EFG measures 110°.

[chart: diagram; triangle DEF with vertices E (top), D (bottom left), F (bottom right), extended side to G; angle labels 45° (at D) and 110° (at F)]

Questions:

1. What is the measure of ∠DFE?
2. Arrange the sides of ∆DEF from shortest to longest.
3. Suppose |DE| = 8 cm, |EF| = 4 cm, |DF| = 11 cm, arrange the angles of ∆DEF from largest to smallest.
4. What are the remote interior angles in the figure?

Activity 2

Refer to each figure below. Write an inequality relating the given pair of angle measures or pair of segment measures.

a.

[chart: diagram; triangle ACE with apex A, base CE containing point R; angle labels 1 and 2 at A on either side of segment AR; segment labels 15 (CR) and 13 (RE); tick marks show congruent sides AC and AE]

m∠1 ☐ m∠2

b.

[chart: diagram; triangle LVE with point O on the segment between L and V, segment OE drawn; angle labels 32° and 35° at E; tick marks show congruent sides LE and VE]

LO ☐ VO

Answer Key

What I Know

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

Assessment

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

What’s In

[transcriber’s note: the printed item 1 of this section is not legible in the image]

2. Triangle Inequality Theorem 1. 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 shorter side.
3. Exterior Angle Inequality Theorem. The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.
4. Triangle Inequality Theorem 2. 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 smaller angle.
5. Converse of Hinge 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.
6. Triangle Inequality Theorem 3. The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

What’s New

Expected Answers:

1. Triangle
2. Yes
3. No, the shorter legs will not meet.

What I Have Learned

A. 1. ∠B 2. Triangle Inequality Theorem 1 3. ∠B, ∠A, ∠C B. 1. 90° 2. shortest $\overline{XY}$; Longest $\overline{XZ}$ 3. Triangle Inequality Theorem 2 C. 1. 6 < x < 16 2. Triangle Inequality Theorem 3 D. 1. ∠BCD 2. 115° 3. a. = ?, b. > [?] E. 1. 95° 2. $\overline{PR}$ 3. Hinge Theorem F. 1. x < 6 [?] 2. Converse of Hinge Theorem

What’s More

Activity 1

1. ∠C, ∠A, ∠B
2. ∠C, ∠B, ∠A
3. ∠C, ∠A, ∠B

Activity 2

A.

1. Yes
2. No
3. No

B.

1. 22 < x < 50
2. 13 < x < 29
3. 27 < x < 53

Activity 3

1. <
2. 
   
3. 
   
4. <
5. =
6. 
   
7. <

Activity 4

1. Leo
2. Hinge Theorem

Activity 5

1. y
2. Use of Converse of Hinge Theorem

What I Can Do

Activity 1

1. 55°
2. ∠M, ∠A, ∠P
3. $\overline{AP}$, $\overline{MP}$, $\overline{AM}$
4. M
5. ∠SPR and ∠MAS
6. ∠MAS

Activity 2

1. No
2. Change
3. [transcriber’s note: the start of this line is cut off in the image] angle, the smaller the angle, the shorter its opposite side

Additional Activities

Activity 1

1. 70°
2. $\overline{DE}$, $\overline{DF}$, $\overline{EF}$
3. ∠D, ∠F, ∠E
4. ∠E, ∠D

Activity 2

a. > b. <

References

Abuzo, Emmanuel P., Bryant, Merden L., Cabrella, Jem Boy B. Caldez, Belen P., Callanta, Melvin M., Castro, Anastacia Preserfina I., Halabaso, Alicia R., Javier, Sonia P., Nocom, Roger T., and Ternida, Conception S. (2013). Grade 8 Mathematics Learners Module. pp. 390-408. Philippines. Book Media Press, Inc. and Printwell, Inc.

Bernabe, Julieta G., et.al. (2009). Revised Edition Geometry, pp. 103-106, 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.

For inquiries or feedback, please write or call:

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Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph