Mathematics 8
Quarter 4 Week 2
NAME: ____________________________________ GRADE & SEC: ________________
Competency:
The learner applies theorems on triangle inequalities. (M8GE-IVb-1).
To the Learners:
Before starting the module, I want you to set aside other tasks that will disturb you while
enjoying the lessons. Read the simple instructions below to successfully enjoy the objectives of this
kit. Have fun!
1. Follow carefully all the contents and instructions indicated in every page of this module.
2. Writing enhances learning. Keep this in mind and take note of the important concepts in your
notebook.
3. Perform all the provided activities in the module.
4. Let your facilitator/guardian assess your answers using the answer key card.
5. Analyze the post-test and apply what you have learned.
6. Enjoy studying!
Expectations
This module is designed to help you master the skills on how to apply theorems on triangle
inequalities as follows:
•
determine the smallest and largest angle of the triangle;
•
write the angles of the triangles in order from smallest to largest;
•
determine the shortest and longest side of the triangle;
•
write the sides of the triangles in order from shortest to longest;
•
determine if a triangle can be formed with the given side length;
•
find the range of the possible measures for the angles and sides using the Triangle
Inequality Theorem; and
•
compare the unequal relationship between side and angle measures.
After going through this module, you are expected to:
1. determine possible measures for the angles and sides of triangles; and
2. justify claims about the unequal relationship between side and angle measures.
DR. CHERRY ANN E. FRANCISCO AND MARIA HERMALYN L. BUHAIN
MATH 8 QUARTER 4 WEEK 2
P a g e 1 | 11
Pre-test
Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper.
1. Which side of ∆ 𝑀𝐴𝑇 is the shortest?
̅̅̅̅̅
A. 𝑀𝐴
̅̅̅̅̅
C. 𝑀𝑇
̅̅̅̅
B. 𝐴𝑇
D. it cannot be determined
2. In ∆RUN, RU = 24 cm, UN = 8 cm, and RN = 30 cm. List the angles in order from
largest to smallest measure.
A. ∠ 𝑅, ∠ 𝑈, ∠ 𝑁
B. ∠ 𝑈, ∠ 𝑅, ∠ 𝑁
C. ∠ 𝑈, ∠ 𝑁, ∠ 𝑅
D. ∠ 𝑁, ∠ 𝑅, ∠ 𝑈
C. 9
D. 7
c. 12< x < 39
d. 12< x < 50
3. What is the possible value of x?
A. 11
B. 10
4. What is the range of the possible value of x?
a. -12< x < 39
b. -15< x < 39
5. What is the range of possible value for x?
a. -4< x < 16
c. -4< x < 20
b. 4< x < 16
d. -10< x < 20
Looking Back to your Lesson
AGAIN!! WHICH IS WHICH?
Directions: Given the following theorems on triangle inequalities in one triangle and triangle
inequalities in two triangles,
a. Triangle Inequality theorem 1 (SS→AA)
b. Triangle Inequality theorem 2 (AA→SS)
c. Triangle Inequality theorem 3 (S1+S2 > S3)
d. Exterior-Angle Inequality Theorem (Exterior ∠ > Remote Interior ∠)
e. Hinge Theorem or SAS inequality Theorem
f. Converse of Hinge Theorem or SSS inequality Theorem
Identify what corresponding illustration for each of the triangle inequalities.
_____1. The sum of the lengths of any two sides of a triangle is greater than the length of the third
side.
_____2. If one angle of a triangle is larger than a second angle, then the side opposite the first angle
is longer than the side opposite the second angle.
_____3. 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.
_____4. 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.
_____5. The measure of an exterior angle of a triangle is greater than the measure of either remote
interior angle.
_____6. If two sides of one triangle are congruent to two sides of another triangle, but the third side
DR. CHERRY ANN E. FRANCISCO AND MARIA HERMALYN L. BUHAIN
MATH 8 QUARTER 4 WEEK 2
P a g e 2 | 11
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.
Introduction of the Topic
Lesson 1: Applies Theorems on Triangle Inequalities
PROPER CONSTRUCTION IS IMPORTANT!!!
In a construction site, you are the engineer who is
going to determine if the constructed triangular bases of
a bridge are identically balance and you observed that
some of the triangular bases look different. How can you
prove that your observation is right?
Based on the situation, application on triangle
inequalities will be utilized for you to prove what you
have observed. Balance and proportion are very
This Photo by Unknown Author is licensed under CC BYSA-NC
important in the construction industry, for if you will
not consider these things accident may happen or worse
can cause death to the people who utilize the constructed things.
Applying concepts on theorems on triangle inequalities (Triangle Inequality Theorem 1,
Triangle Inequality Theorem 2, Triangle Inequality Theorem 3, Exterior-Angle Inequality Theorem,
Hinge Theorem or SAS Inequality Theorem, and Converse of Hinge Theorem or SSS Inequality
Theorem) are very useful in checking properties of triangle-shaped objects. Again, here are the
theorems on triangle inequalities.
Inequalities in One Triangle:
1. Triangle Inequality Theorem 1 (Ss → 𝑨𝒂)
Example 1: Cite angles and sides relationship shown in the given triangle.
̅̅̅̅
𝐴𝐵 > ̅̅̅̅
𝐵𝐶 , So m∠ 𝐶 > m∠ 𝐴
Example 2: Name the smallest and largest angle of the triangle.
Largest angle: ∠𝑀
Smallest angle: ∠𝐾
Example 3: Write the angles of the triangles from smallest to largest order.
̅̅̅̅
𝐵𝐶 < ̅̅̅̅
𝐴𝐵 < ̅̅̅̅
𝐴𝐶 , So m∠ 𝐴 < m∠ 𝐶 < m∠ 𝐵
Angle: ∠ 𝐴, ∠ 𝐶, ∠ 𝐵
2. Triangle Inequality Theorem 2 (Aa → 𝑺𝒔)
Example 1: Name the shortest and longest side of the triangle.
Since the m∠𝐵 = 1000, m∠𝐶 = 500, and m∠𝐴 = 300, therefore the
shortest side is ̅̅̅̅
𝐵𝐶 and the longest side is ̅̅̅̅
𝐴𝐶 .
Example 2: Write the sides of the triangle from shortest to longest measure.
Solution:
Since 𝑚∠ 𝐸= 610 and 𝑚∠ 𝐹= 590,
By the Triangle Sum Theorem: find 𝑚∠ 𝐷
DR. CHERRY ANN E. FRANCISCO AND MARIA HERMALYN L. BUHAIN
MATH 8 QUARTER 4 WEEK 2
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m∠ 𝐷 = 1800 – (𝑚∠ 𝐸 + 𝑚∠ 𝐹)
m∠ 𝐷 = 1800 – (610 + 590)
m∠ 𝐷 = 1800 – (1200) = 600
𝑚∠ 𝐹 < 𝑚∠ 𝐷 < 𝑚∠ 𝐸, So ̅̅̅̅
𝐷𝐸 < ̅̅̅̅
𝐸𝐹 < ̅̅̅̅
𝐷𝐹
Side: ̅̅̅̅
𝐷𝐸, ̅̅̅̅
𝐸𝐹 , ̅̅̅̅
𝐷𝐹
Example 3: List the angles and sides of each triangle from smallest/shortest to
largest/longest measure.
Solution:
Using the Triangle Sum Theorem, we can solve for x, as shown below.
(2x + 9) + (2x + 1) + 90 = 180
Triangle Sum Theorem
4x + 100 = 180
combine like terms
4x = 80
APE, add (– 100) to both sides
x = 20
MPE, divide both sides by 4
m∠𝑋 = 2 (20) + 1 = 410, and m∠𝑌 = 2 (20) + 9 = 490. Therefore,
m∠𝑋 < m∠𝑌 < m∠𝑍, we know that the length of sides across larger angles are
longer than those across from shorter angles so ̅̅̅̅
𝑌𝑍 < ̅̅̅̅
𝑋𝑍 < ̅̅̅̅
𝑋𝑌 .
̅̅̅̅, 𝑋𝑍
̅̅̅̅, 𝑋𝑌
̅̅̅̅
Side: 𝑌𝑍
Angle: ∠ 𝑋, ∠ 𝑌, ∠ 𝑍
3. Triangle Inequality Theorem 3 (S1 + S2 > S3)
Example 1: Determine if a triangle can be formed with the given side length.
Explain your answer.
a. 4, 8, 10
Solution:
4+8
?
10
>
4 + 10
12 > 10 True
?
8
>
8 + 10
14 > 8 True
?
4
>
18 > 4 True
Conclusion: The sum of each pair of side lengths is greater than the third
length. So, a triangle can have side lengths of 4, 8, and 10.
b. 7, 9, 18
Solution:
7+9
?
18
>
7 + 18
16 > 18 False
?
9
>
9 + 18
25 > 9 True
?
7
>
27 > 7 True
Conclusion: Not all three inequalities are true. So, a triangle cannot have these
three side lengths.
Example 2: Find the range of values for x using the Triangle Inequality Theorem.
Solution:
x + 14 > 21
x>7
21 + 14 > x
x + 21 > 14
35 > x
x > -7
Ignore the inequality with a negative value since a triangle cannot have a
negative side length. Combine the inequalities. So, the length of the third side is
greater than 7 and less than 35, (7 < x < 35)
Example 3: Find the range of values for x using the Triangle Inequality Theorem.
Solution:
DR. CHERRY ANN E. FRANCISCO AND MARIA HERMALYN L. BUHAIN
MATH 8 QUARTER 4 WEEK 2
P a g e 4 | 11
(x + 2) + 10 > 12
(x + 2) + 12 > 10
10 + 12 > x + 2
x + 12 > 12
x + 14 > 10
22 > x + 2
x>0
x > -4
20 > x
Combine the inequalities. Hence, 0 < x < 20. The length of the third side
is greater than 0 and less than 20.
4. Exterior Angle Inequality Theorem
Use the Exterior Angle Inequality Theorem to answer the following.
Example 1: Show angles that is less than m∠ 4
Solution:
By the Exterior Angle Inequality Theorem, the exterior angle ∠ 4 is larger than
either the two remote interior angles (∠ 1 and ∠ 2). Also, m∠ 4 + m∠ 3 = 180, and
(m∠ 1 + m∠ 2) + m∠ 3 = 180. By transitivity, m∠ 4 = (m∠ 1 + m∠ 2). Therefore, ∠ 4
must be larger than each individual angle. By the Exterior Angle Inequality
Theorem, m∠ 𝟒 > 𝑚∠ 1 and m∠ 𝟒 > 𝒎∠ 𝟐.
Answer: ∠ 𝟏, ∠ 𝟐
Example 2: Find the possible value of x.
Solution: Find x
∠ 𝑥 > 500; ∠ 𝑥 > 600, however you know x = sum of the remote interior angle.
Therefore, 600 < x < 1800.
Example 3: Find the restrictions on x.
Solution:
30 < 3x – 18 EAI Theorem
3x – 18 < 180
48 < 3x
add both sides by 18
3x < 198
add both sides by 18
16 < x
divide both sides by 3
x < 66
divide both sides by 3
Therefore, 16 < x < 66.
Inequalities in Two Triangles:
1. Hinge Theorem or SAS Inequality Theorem
̅̅̅̅ and 𝑌𝑍
̅̅̅̅.
Example 1: Compare 𝐵𝐶
Solution:
̅̅̅̅ ≅ 𝑋𝑌
̅̅̅̅, 𝐴𝐶
̅̅̅̅ ≅ 𝑋𝑍
̅̅̅̅ and 𝑚∠ 𝐵𝐴𝐶 > 𝑚∠𝑌𝑋𝑍; then 𝑩𝑪
̅̅̅̅ > 𝒀𝒁
̅̅̅̅.
If 𝐴𝐵
Example 2: Compare ̅̅̅̅
𝐸𝐹 and ̅̅̅̅
𝐺𝐹.
Solution:
Compare the sides and angles in ∆ 𝐸𝐹𝐻 angles in ∆ GFH.
m∠𝐺𝐻𝐹 = 1800 – 820 = 980.
̅̅̅̅
𝐸𝐻 = ̅̅̅̅
𝐺𝐻
̅̅̅̅
𝐹𝐻 = ̅̅̅̅
𝐹𝐻
𝑚∠ 𝐸𝐻𝐹 > 𝑚∠𝐺𝐻𝐹.
By the Hinge Theorem, ̅̅̅̅
𝑬𝑭 < ̅̅̅̅̅
𝑮𝑭.
Example 3: Solve for the possible values of x.
Solution:
x + 23 > 3x + 9 (theorem)
-2x > -14
(combine like terms)
DR. CHERRY ANN E. FRANCISCO AND MARIA HERMALYN L. BUHAIN
MATH 8 QUARTER 4 WEEK 2
3x + 9 > 0
3x > -9 (add both sides by -9)
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x< 7
(divide both sides by -2)
x > -3 (divide both sides by 3)
Therefore, -3 < x < 7.
2. Converse of Hinge Theorem or SSS Inequality Theorem
Example 1: Compare 𝑚∠ 𝐵𝐴𝐶 and 𝑚∠𝐷𝐴𝐶
Solution:
Compare the side length in ∆ 𝐴𝐵𝐶 and ∆ ADC.
̅̅̅̅
𝐴𝐵 = ̅̅̅̅
𝐴𝐷
̅̅̅̅
𝐴𝐶 = ̅̅̅̅
𝐴𝐶
̅̅̅̅
𝐵𝐶 > ̅̅̅̅̅
𝐷𝐶
By the Converse of the Hinge Theorem, m∠𝐵𝐴𝐶 > 𝑚∠𝐷𝐴𝐶.
Example 2: Find the range of values for k.
Solution:
Step 1: Compare the side length in ∆ 𝑀𝐿𝑁 and ∆ PLN.
̅̅̅̅
𝐿𝑁 = ̅̅̅̅
𝐿𝑁
̅̅̅̅ = ̅̅̅̅
𝐿𝑀
𝐿𝑃
̅̅̅̅̅ > ̅̅̅̅̅
𝑀𝑁
𝑃𝑁
By the Converse of the Hinge Theorem, m∠𝑀𝐿𝑁 > 𝑚∠𝑃𝐿𝑁.
5k – 12 < 38
k < 10
(substitute the given values)
(add 12 to both sides and divide by 5)
Step 2: Since ∠𝑃𝐿𝑁 is in a triangle, m∠𝑃𝐿𝑁 > 00.
5k – 12 > 0
k < 2.4
(substitute the given values)
(add 12 to both sides and divide by 5)
Step 3: Combine the two inequalities. The range of values for k is 2.4 < k < 10.
Activities
ACTIVITY 1: MEASURES MATTERS!
Directions: Given the illustrated triangles and corresponding measurements,
A. Identify the largest angle and the smallest angle then write the angles of the triangles from
smallest to largest order.
Triangle
1
∆ABC
2
∆DEF
3
∆CAB
Largest
Smallest
Angle
Angle
Order of angle measures from smallest to largest:
1. ____________________
2. ____________________
3. ___________________
B. Identify the longest side and the shortest side then write the sides of the triangles in order from
shortest to longest.
Triangle
1
2
3
Longest
Side
∆ABC
∆LMN
∆DEF
Order of side lengths from shortest to longest:
DR. CHERRY ANN E. FRANCISCO AND MARIA HERMALYN L. BUHAIN
MATH 8 QUARTER 4 WEEK 2
P a g e 6 | 11
Shortest
Side
1. ____________________
2. ____________________
3. ___________________
C. RAMPS. The wedge at the right represents a skateboard ramp. Which is
̅̅̅̅?
longer, the length of the ramp ̅̅̅̅
𝐾𝐴 or the length of the top surface of ramp 𝑆𝐴
Explain your answer. ________________________________________________________
ACTIVITY 2: A TRIANGLE OR NOT?
A. Determine whether the given measures can be the lengths of the sides of a
triangle. Write Yes or No.
_______ 1. 5, 4, 3
_______ 6. 12, 15, 18
_______ 2. 5, 15, 10
_______ 7. 13, 14, 29
_______ 3. 30.1, 0.8, 31
_______ 8. 17, 30, 30
_______ 4. 8.4, 7.2, 3.5
_______ 9. 9, 21, 20
_______ 5. 5, 17, 9
_______10. 18, 32, 21
B. Find the range of values for x using the Triangle Inequality Theorem.
_____________ 1. 7, 12 and x
_____________ 4. 15, 18 and x
_____________ 2. 14, 23 and x
_____________ 5. 10, 14 and 2x + 2
_____________ 3. 22, 34 and x
C. Mrs. Barreto has a pet rabbit and wants to build a pen for it. She has 3 pieces of
lumber: one is 3 ft, one is 7 ft, and the other is 8 ft long. Can she build a closed
triangular pen with these three lumbers (will the lumbers form a triangle)? Explain your answer.
________________________
_______________________________________________________________________________________________
ACTIVITY 3: GREATEST OF THEM ALL!
Determine which angle has the greatest measure. Write your answer on the space provided.
_______ 1. ∠ 1, ∠ 2, ∠ 4
_______ 4. ∠ 5, ∠ 7, ∠ 8
_______ 2. ∠ 1, ∠ 2, ∠ 6
_______ 5. ∠ 5, ∠ 6, ∠ 8
_______ 3. ∠ 3, ∠ 5, ∠ 7
ACTIVITY 4: FINDING “x”.
Find the range of the possible values for x.
1.
2.
_________________
3.
_________________
_________________
ACTIVITY 5: COMPARE US!
Write an inequality relating the given pair of angles or segment measures. Complete each statement
by writing <, =, or >.
̅̅̅̅ ______ 𝑃𝑁
̅̅̅̅
1. 𝐿𝑀
̅̅̅̅ ______ 𝐷𝐶
̅̅̅̅
2. 𝐴𝐷
DR. CHERRY ANN E. FRANCISCO AND MARIA HERMALYN L. BUHAIN
MATH 8 QUARTER 4 WEEK 2
̅̅̅̅ ______ 𝑁𝐾
̅̅̅̅̅
3. 𝑆𝑇
̅̅̅̅ ______ 𝐼𝐽
̅
4. 𝐻𝐼
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5. 𝑚∠𝐶𝐴𝐵 ______ 𝑚∠𝐶𝐵𝐴
6. 𝑚∠1 ______ 𝑚∠2
7. 𝑚∠1 ______m∠2
8. 𝑚∠𝐴 _____m∠𝐵
ACTIVITY 6: SOLVE ME!
Use the Hinge Theorem or its converse and properties of triangle to write and solve an inequality to
describe the restriction on the value of x.
1.
2.
3.
4.
5.
Remember
You have learned the following theorems on Triangle inequalities:
Inequalities in One Triangle:
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
Inequalities in Two Triangles:
1. Hinge Theorem or SAS Inequality Theorem
2. Converse of Hinge Theorem or SSS Inequality Theorem
Check your Understanding
Directions: Identify the theorem to be applied in the given situations/problems, then solve if
necessary. Write your answer on the space provided.
1. If the measures of the two sides a triangle is 15 and 26, what are the possible measures of the
third side? __________________________________
2. If AB = 7cm, BC = 8cm and AC = 13cm. Which among the angles is the largest? _________________
3. If the measures of the two angles of ∆KIT are ∠K = 75˚ and ∠I = 32˚, what is the longest side?
__________________________________
4. List the sides of ∆PQR in order from longest to smallest if the angles of ∆PQR have the given
measures: m∠𝑃 = 7x + 8, m∠𝑄 = 8x – 10, m∠𝑅 = 7x + 6. ______________________________
5. The side lengths of ∆NFL are NF = 17, FL = 19, and NL = 10x - 11. Determine the possible
DR. CHERRY ANN E. FRANCISCO AND MARIA HERMALYN L. BUHAIN
MATH 8 QUARTER 4 WEEK 2
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values of x. __________________________________
6. Using the figure and the Inequality Theorem, which angle, ∠5 or ∠8, has the
smallest measure? _____________________________________
7. Using the figure and the Inequality Theorem, which angle, ∠1, ∠6 or ∠9, has
the greatest measure? ____________________________________
Use the figure at the right.
̅̅̅̅ = 𝐸𝐻
̅̅̅̅ and m∠FEG > m∠FEH, which is longer, 𝐻𝐹
̅̅̅̅ or 𝐺𝐹
̅̅̅̅ ? ____________________
8. If 𝐸𝐺
̅̅̅̅ = 𝐸𝐻
̅̅̅̅ and 𝐺𝐹
̅̅̅̅ < 𝐻𝐹
̅̅̅̅ , which is larger, ∠GEF or ∠HEF? ________________________
9. If 𝐸𝐺
10. Solve an inequality to describe a restriction on the value of x. ______________________
Post-Test
Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper.
1. Which side of ∆ 𝑀𝐴𝑇 is the shortest?
a. ̅̅̅̅̅
𝑀𝐴
c. ̅̅̅̅̅
𝑀𝑇
b. ̅̅̅̅
𝐴𝑇
d. it cannot be determined
2. In ∆RUN, RU = 24 cm, UN = 8 cm, and RN = 30 cm. List the angles in order from
smallest to largest measure.
a. ∠ 𝑅, ∠ 𝑈, ∠ 𝑁
b. ∠ 𝑈, ∠ 𝑅, ∠ 𝑁
c. ∠ 𝑈, ∠ 𝑁, ∠ 𝑅
d. ∠ 𝑁, ∠ 𝑅, ∠ 𝑈
c. 9
d. 7
c. 12< x < 39
d. 12< x < 50
3. What is the possible value of x?
a. 11
b. 10
4. What is the range of the possible value of x?
a. -12< x < 39
b. -15< x < 39
5. What is the range of possible value for x?
a. -4< x < 16
c. -4< x < 20
b. 4< x < 16
d. -10< x < 20
Additional Activities
To better understand the lesson, watch the video lesson on https://www.youtube.com/watch?v=3vOfQnUjzV8
entitled “Triangle Inequality Theorem” and https://www.youtube.com/watch?v=YHwSe-0c7Xo entitled “Hinge
Theorem Inequalities 2 Triangles”.
Reflection
At the beginning of the lesson, I couldn’t _______________________________
_______________________________________________________________________
But now I can _________________________________________________________
_______________________________________________________________________
Here’s how I did it: ___________________________________________________
_______________________________________________________________________
What this means for my learning next year? ____________________________
_______________________________________________________________________
DR. CHERRY ANN E. FRANCISCO AND MARIA HERMALYN L. BUHAIN
MATH 8 QUARTER 4 WEEK 2
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