Republic of the Philippines Department of Education Regional Office IX, Zamboanga Peninsula Zest for Progress Zeal of Partnership 8 4 th QUARTER – Module 2: APPLICATIONS OF TRIANGLE INEQUALITIES Name of Learner: ___________________________ Grade & Section: ___________________________ Name of School: ___________________________
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Republic of the Philippines
Department of Education Regional Office IX, Zamboanga Peninsula
Zest for Progress
Zeal of Partnership
8
4th QUARTER – Module 2: APPLICATIONS OF TRIANGLE
INEQUALITIES
Name of Learner: ___________________________
Grade & Section: ___________________________
Name of School: ___________________________
1
Mathematics – Grade 8 Alternative Delivery Mode Quarter 4 - Module 2: Applications of Triangle Inequality First Edition, 2020
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𝑏 + 𝑐 > 𝑎 4 + 10 > 3 or 14 > 3 (True) Since one of the situations is false, therefore, we can
conclude that we cannot form a triangle with side lengths 3, 4, and 10 units.
D.
∠ACD=1000
E. In ∆𝑋𝑌𝑍 , if 𝑋𝑍̅̅ ̅̅ > 𝑌𝑍̅̅̅̅ and
𝑌𝑍̅̅̅̅ > 𝑋𝑌̅̅ ̅̅ , then the following mathematical statements are true.
𝑚∠𝑌> 𝑚∠𝑋
𝑚∠𝑋> 𝑚∠𝑍
𝑚∠𝑌> 𝑚∠𝑍
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What’s New
Mang Juan is fencing the triangular lot that he was able
to purchase. His available material is 11-meter barbed wire.
How much more barbed wire will he need to fence his lot?
What is It
ILLUSTRATIVE EXAMPLE 1
Use the Exterior Angle Inequality theorem to write the inequalities observable in the figure shown.
4 meters
7 meters
Based on the activity, how did you come up with the measurement of the third side of the triangular garden?
In this module, we will apply the triangle inequality theorems in different kinds of problem.
The measure of an exterior angle of a triangle is greater than the measure of any of its remote interior angles.
Exterior Angle Inequality Theorem
SOLUTION Consider ∆𝐻𝐴𝑇:
𝑚∠𝐻𝐴𝑀 > 𝑚∠𝐻𝑇𝐴
𝑚∠𝐻𝐴𝑀 > 𝑚∠𝐴𝐻𝑇
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ILLUSTRATIVE EXAMPLE 2
Given that the measure of the exterior angle is equal to the sum of the measures of its
remote interior angles, solve for 𝑥.
ILLUSTRATIVE EXAMPLE 1
In ∆𝑋𝑌𝑍, if 𝑋𝑍̅̅ ̅̅ > 𝑌𝑍̅̅̅̅ and 𝑌𝑍̅̅̅̅ > 𝑋𝑌̅̅ ̅̅ , then it follows that,
𝑚∠𝑌 > 𝑚∠𝑋 𝑚∠𝑋 > 𝑚∠𝑍 𝑚∠𝑌 > 𝑚∠𝑍
ILLUSTRATIVE EXAMPLE 2 Given ∆𝐺𝐻𝐼 whose perimeter is 50 units, what is the largest angle?
SOLUTION:
Since the longest side is 𝐻𝐼, therefore the angle opposite to it is the largest angle
which is 𝐺.
𝑚𝐴 + 𝑚𝐵 = 𝑚𝐴𝐶𝐷
(2𝑥 + 3)° + (3𝑥 + 2)° = 135°
2𝑥 + 3 + 3𝑥 + 2 = 135
5𝑥 + 5 = 135
5𝑥 = 135 − 5
5𝑥 = 130
5𝑥
5=
130
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𝑥 = 26°
First, solve for 𝑥. 𝐺𝐼 + 𝐻𝐼 + 𝐺𝐻 = 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟
2𝑥 + 1 + 4𝑥 − 2 + 𝑥 + 2 = 50
7𝑥 + 1 = 50
7𝑥 = 50 − 1
7𝑥 = 49
7𝑥
7=
49
7
𝑥 = 7 𝑢𝑛𝑖𝑡𝑠
Second, find the measure of each side.
𝐺𝐼 = 2𝑥 + 1 = 2(7) + 1 = 15𝑢𝑛𝑖𝑡𝑠
𝐻𝐼 = 4𝑥 − 2 = 4(7) − 2 = 26 𝑢𝑛𝑖𝑡𝑠
𝐺𝐻 = 𝑥 + 2 = 7 + 2 = 9 𝑢𝑛𝑖𝑡𝑠
If one side of a triangle is longer than the second side, then the angle
opposite the first side is larger than the angle opposite the second side.
Triangle Inequality Theorem 1 (Ss → Aa)
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ILLUSTRATIVE EXAMPLE 1
Name the shortest side and the longest side of the following triangles.
Triangle Longest Side Shortest Side
1 ∆𝐽𝐾𝐿 𝐽𝐾̅̅ ̅ 𝐽�̅�
2 ∆𝑀𝑁𝑂 𝑀𝑂̅̅ ̅̅ ̅ 𝑀𝑁̅̅ ̅̅ ̅
3 ∆𝑃𝑄𝑅 𝑃𝑅̅̅ ̅̅ 𝑄𝑅̅̅ ̅̅
ILLUSTRATIVE EXAMPLE 2 What is the shortest side of ∆𝐽𝐾𝐿?
To find the shortest side, solve first for 𝑥. 𝑚𝐽 + 𝑚𝐾 + 𝑚𝐿 = 180°
(3𝑥)° + (2𝑥 − 5)° + (𝑥 − 7)° = 180°
3𝑥 + 2𝑥 − 5 + 𝑥 − 7 = 180
6𝑥 − 12 = 180
6𝑥 = 180 + 12
6𝑥 = 192
6𝑥
6=
192
6
𝑥 = 32°
Next, find the measure of each angle.
𝑚𝐽 = 3𝑥 = 3(32) = 96°
𝑚𝐾 = 2𝑥 − 5 = 2(32) − 5 = 59°
𝑚𝐿 = 𝑥 − 7 = 32 − 7 = 25°
Since the smallest angle is 𝐿 ,then the shortest
side is 𝐽𝐾.
If one angle of a triangle is larger than the second angle, then the side opposite the first angle is longer than the side opposite the second angle.
Triangle Inequality Theorem 2 (Aa → Ss)
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Triangle Inequality Theorem 3 (S1 + S2 > S3)
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ILLUSTRATIVE EXAMPLE If two sides of a triangle have lengths 7 cm and 10 cm, what are the possible lengths of the third side?
If t is the third side then, the following should be satisfied:
Inequality 1 Inequality 2 Inequality 3
7 + 10 > t
t < 7+10
t < 17
t must be less than 17
7 + t > 10
t > 10 − 7
t > 3
t must be greater than 3
t + 10 > 7 t > 7 − 10
t > -3
Nonpositive values of t to be disregarded.
The resulting inequalities show that t must be between 3 and 17. Side t may have the following measurements (in cm): {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}.
The inequality model representing the possible length of the third side is 𝟑 < 𝒕 < 𝟏𝟕.
ILLUSTRATIVE EXAMPLES Application 1. Using Hinge Theorem and its Converse, solve for the possible values of m.
SOLUTION:
2𝑚 − 1 > 𝑚 + 4 2𝑚 − 𝑚 > 1 + 4
𝑚 > 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.
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.
Converse of Hinge Theorem (SSS Triangle Inequality Theorem)
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Therefore, Adee is farther from the center of the plaza than Ella.
2. Ella and Adee enjoy biking every afternoon at the plaza. One day, they decided to go on
different directions. From the center of the plaza, Ella bikes 40 meters east and then 50
meters south. Adee bikes 50 meters west. He then takes a right of 70° and bikes 40 meters.
Who is farther from the center of the plaza?
SOLUTION:
What’s More
Directions: Write your answers on a separate sheet.
A. Given the following measurements of the sides of a triangle, list down its angles from smallest to largest.
1. 𝐴𝐵 = 17 , 𝐵𝐶 = 21, 𝐴𝐶 = 18
2. 𝐴𝐵 = 15 , 𝐴𝐶 = 16, 𝐵𝐶 = 17 B. Given the following measurements of the angles of a triangle, list down its sides from
C. Decide whether each set of numbers can form a triangle.
Given a + b > c a + c > b b + c > a Triangle? YES/NO
EXAMPLE: {5,8,10}
13>10 15>8 18>5 YES
1. {5, 5, 3}
2. {6, 2, 7}
3. {11, 8, 3}
4. {5, 5, 10}
5. {4, 5, 8}
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D. Using Hinge Theorem and Its Converse, write a conclusion about each statement. Use the symbols >, < or = to complete the statements about the figure shown below.
What I Have Learned
Directions: Briefly answer the questions below.
QUESTIONS RESPONSES
1. How would you compare the measure of the exterior angle and its remote interior angles?
2. Where would the largest angle of a triangle be located? (Justify your answer using the Triangle inequality theorem 1.)
3. Where would the shortest side of a triangle be located? (Justify your answer using Triangle Inequality Theorem 2)
4. Can the following side measures form a triangle? 2,2,10. Why?
5. In figure 1, ∠𝐷𝐴𝐶 is the exterior angle
while ∠𝐴𝐵𝐶 and ∠𝐴𝐶𝐵 are the remote interior angles. Describe the measurements of the exterior angle and the remote interior angles.
What I Can Do
Directions: Briefly answer the reflection questions below.
1. What is the importance of learning Triangle Inequality Theorems?
2. How can you apply the Triangle Inequality Theorems in real-life problems?
Direction: Read each item carefully. Write the letter of the correct answer on a separate sheet. 1. Triangle inequality theorem 3 states that the sum of the lengths of any two sides of a
triangle is _______ the length of the third side.
A. greater than B. less than
C. equal to D. greater than but not equal to
2. Which of the following statements is correct?
A. The sum of the remote interior angle is equal to the measurement of the exterior angle
of the triangle.
B. The sum of the remote interior angle is greater than the measurement of the exterior
angle of the triangle.
C. The sum of the remote interior angle is less than the measurement of the exterior angle
of the triangle.
D. The measurements of the angles of an equilateral triangle are not equal.
3. The largest angle of a triangle is located across the _________ of the triangle.
A. from the smallest side B. from the longest side
C. adjacent to the smallest side D. adjacent to the longest side
4. Where would the longest side of a triangle be located?
A. Across from the smallest angle of a triangle
B. Adjacent to the largest side of a triangle
C. Across the largest angle of a triangle
D. In the middle of a triangle
5. Which of these lengths CAN represent the sides of a triangle?
I. 3 meters, 4 meters, 5 meters
II. 2 inches, 2 inches, 10 inches
III. 5 feet, 5 feet, 5 feet
IV. 10 meters, 3 meters, 2 meters
A. I, III, and IV B. I and II C. I and III D. III only
6. The exterior angle of a triangle measures 100o. if one of the remote interior angles of the
same triangle is 58o, what is the measurement of the other remote interior angle?
A. 42° B. 43° C. 45° D. 92°
7. What is the measurement of the exterior angle of an equiangular triangle?
A. 120° B. 100° C. 60° D. 30°
8. Which of the following does not represent the lengths of the sides of a triangle?
A. 2cm,2cm,1cm B. 3cm,3cm,15cm
C. 5cm,4cm,3cm D. 1cm,1cm,1cm
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9. Given the lengths of two sides of a triangle. Find the range of lengths of the third side: 13
cm and 7 cm.
A. 6 < 𝑥 < 20 B. 20 < 𝑥 < 6
C. 6 < 𝑥 > 20 D. 6 > 𝑥 > 20
10. Hinge Theorem states that” If two triangles have two congruent sides, then the one with the larger included angle has the ____________”.
A. longer third side B. shorter third side
C. largest angle D. shortest angle
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Answer Key
What’s In:
1.D 2. E 3. B 4. C 5. A
What’s New:
3 meters to 6 meters
What I Know:
1.B 2. D 3. D 4. A 5. A
What’s More:
A. B.
1. smallest- C 1. smallest- 𝑨𝑪̅̅̅̅ 3. smallest- 𝑩𝑪̅̅̅̅
larger - B longer - 𝑪𝑩̅̅̅̅ longer - 𝑨𝑪̅̅̅̅
largest - A longest - 𝑨𝑩̅̅̅̅ longest - 𝑨𝑩̅̅̅̅
2. smallest- C 2. smallest- 𝑩𝑪̅̅̅̅
larger - B longer - 𝑨𝑪̅̅̅̅
largest - A longest - 𝑨𝑩̅̅̅̅
What’s More:
C. D.
Given a + b > c a + c > b b + c > a Triangle? Yes/No
1 5,5,3 10> 8>5 8>5 YES
2 6,2,7 8>7 13>2 9>6 YES
3 11,8,3 19>3 14>8 11=11 NO
4 5,5,10 10=10 15>5 15>5 NO
5 4,5,8 9>8 12>5 13>4 YES
𝐵𝐶̅̅̅̅ = 𝐵𝐷̅̅̅̅
𝑚∠4 = 𝑚∠3
𝐵𝐷̅̅̅̅ < 𝐵𝐶̅̅̅̅
𝑚∠3 > 𝑚∠4
What I Have Learned:
(For Item 1-4, answers may vary.)
1.The measure of the exterior angle is greater than the measure of the remote interior
angle.
2.The largest angle is opposite the longest side.
3.The shortest side is opposite the smallest angle.
4.No, because one condition will not be satisfied (2+2<10).
5.𝑚𝐷𝐴𝐶>𝑚𝐴𝐵𝐶
𝑚𝐷𝐴𝐶>𝑚𝐴𝐶𝐵
What I Can Do: (Answers may vary.)
1.It can help us solve some real-life problems. 2.It can be used in architecture, engineering, fashion trends and the like.