for Learners

Mathematics 8
Fourth Quarter, Week 2
Objective:
.Applies theorems on triangle inequalities

MELC Code:
M8GE-IVb-1

Written by:

Geraldine C. Dela Cerna

Kabasalan Science and Technology High School

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A. Mini Lesson:
"Any side of a triangle is less than the sum of the other two sides." (Triangle
Inequality Theorem)
In equation form, the Triangle Inequality Theorem looks like this:
a < b+c b¿a+c c¿a+b

The Triangle Inequality Theorem is defined as “The sum of the length of the
two sides of a triangle is greater than the length of the third side”. This
theorem helps determine if any three lengths can create a triangle.
Let’s determine if it’s possible to draw a triangle with following measures.
5, 10, 12

5 + 10 ¿ 12
10 + 12 ¿ 5
5 10
12 + 5 ¿ 10

12
6, 7, 8 6 +7 ¿ 8
7+8 ¿ 6
The sum of
8 +6 ¿ 7 the two
7 smaller
6
numbers is
greater than
the larger
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“The sum of the length of the two sides of a triangle is greater than the length of
the third side.”
Let’s determine if it’s possible to draw a triangle with following measures below.
6, 6, 12
6 + 6 ≯12 X
6 + 12 ¿ 6
6
6 12 + 6 ¿ 6

Since 6 + 6 ≯ 1212, therefore it is not possible to draw a triangle with

measures 6, 6, and 12.

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If Third St. is one of the If the Third St. is the
shorter streets. longest street.
0.75 + x ¿ 1.25 0.75 + 1.25 ¿ x
0.50 ¿ x 2.00 ¿ x

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Example 2: If the measures of two sides are 5 yards and 9 yards what is the least
possible measure of the third side if the measure is an integer 5 𝑦𝑑,9 𝑦𝑑, 𝑥 𝑦𝑑 ?
5 +3,94,> 10
𝑥 5+𝑥>9 9+𝑥>5 3 + 4 ≯ 10 X
3 These
14 > 𝑥 𝑥>4 𝑥>4−4 didn’t 4 + 10 ¿ 3
𝑥 < 14 work 10 + 3 ¿ 4
becau
10
se the
Since x has to be an integer that is greater than 4, the least possible
sum
measure of the third side can have a measure of 5.
of the
Since 3 + 4 ≯ 10, therefore itsmalle
is not possible to draw a triangle with the
measures
Example 3: 3, 4, and
Find 10
the range of possible rmeasures
two of 𝑥 if each set of expressions
sides
represents measures of the sides of a triangle. A. 𝑥 + 1 ,5 ,7
were
x+1+5>7 x + 1 + 7 > 5 not 5+7>x+1
x+6>7 x+8>5 greate 12 > x + 1
r than
x>1 x > -3 the 11 > x
larges
The range is 1¿ x ¿ 11. t side.
The Triangle Inequality Theorem
Task:
The sum of the length of any two sides Q
of aistriangle mustabe greater R
John conducting survey in athan the
neighborhood. He has already walked down First
St.Length of the St.
and Second thirdHis
side.
pedometer showed that he walked 0.75 miles on the First
St. and now reads 2.00 miles total. He is about to take P
Third St. to return to where
he started the survey. He doesn’t know if he should call a cab or just walked. Help
John figure out what the possible lengths of the Third St.
PQ + QR ¿ PR; QR+ PR ¿ PQ; PR+ PQ ¿ QR

First St.
Example 1: Is it possible to form
Second St. a triangle withSt.
Can Third thebegiven side2.00
exactly lengths?
miles? Can
it be exactly 0.5 miles? What would be
A. 15 𝑦ar𝑑s, 16 𝑦ar𝑑s, 30 𝑦ar𝑑s
Third St. the triangle look like?

Triangle Inequality Theorem states, that the sum of any two sides of a triangle
15 +
be16 > 30 than15
the+third
30 >side.
16 16 + 30 > 15
must larger
31 > 30 45 > 16 46 > 15

Task
Solution
YES! Since the sum of each pair of side lengths is greater than the third side
length. Side lengths 15 yards, 16 yards and 30 yards will form a triangle.

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How can this figure help us prove the theorem?

A D
B

Extend the segment AB ´ of ∆ ABC to point D. So that BD

´ is congruent to BC.
´ Since
´ is congruent to BC
BD ´ and in the same triangle, then their opposite angles are
also congruent by the theorem stating that in a triangle the larger angle opposite
the larger side, shorter angle is opposite the shorter side, and congruent angles
are opposite congruent sides.
Thus, ∠ ACD is larger than ∠ BCD because the sum of two positive real numbers
is greater than the two addends. Since ∠ ACD is the greater than angle D. So the
´ is greater than AC
AD ´ since the larger angle is opposite the larger side in any one
triangle.
´
AD= ´ B´D∧ BD=
AB+ ´ ,
´ BC
´
AD= ´ BC
AB+ ´ . So , AB+
´ BC ´ by substitution.
´ > AC

Therefore, in any triangle the sum of any two sides must be larger than the third
side of the triangle.

Remember!
You can determine which side of a triangle is the largest or
smallest, by only knowing the angles, and vice versa.
When you know two sides of a triangle, there is only a range of
possibilities for the third side.
You can determine if three segments can make a triangle.

In ∆ ABC, A = 400 , B = 800 , C = 600. List the sides in order from least to
greatest.

B
BC is the shortest side as it is opposite to angle A
80°
AC is longest side as it is opposite to angle B
60°
A Therefore:
40°
´ < AB<
BC ´ AC
´
C

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Okay let’s try another example
Find the value of x and list the sides of triangle KLM in order from shortest to
longest if the angle have indicated measures. So we have,
m∠ A = ( 12x- 9)0 , m∠ B = (62-3x)0 and m∠C = ( 16x+ 2)0

m∠ A + m∠ B + m∠C = 1800
C
12x – 9 + 62 – 3x + 16x +2 = 1800
25x + 55 = 1800 820

25x = 180- 55
25 x 125
25
= 25 550 A
470
x=5
B

A = 12x – 9 B = 62- 3X C = 16x + 2

= 12(5) - 9 = 62 – 3(5) = 16(5) + 2
= 60 - 9 = 62 – 15 = 80 + 2
A = 550 B = 470 C= 820
The order of the sides from shortest to greatest is AC
´ , BC
´ , AB.
´

Our next example is about discovering the triangle inequality theorem

involving an exterior angle of a triangle. Before doing it, let us first recall the
definition of an exterior angle of a triangle.
L

M N P
By extending MN of ∆LMN to a point P, MP ´ is formed. As a result, ∠LNP forms a
linear pair with ∠LNM. Because it forms a linear pair with one of the angles of ∆
LMN, ∠LNP is referred to as an exterior angle of ∆LMN. The angles non-
adjacent to ∠LNP, ∠L and ∠M, are called remote interior angles of exterior ∠
LNP.
Now suppose a triangle has three sides with lengths a, b, and c.
Exterior
Remote 2 angle
m∠4 = m∠1 + m∠2
interior
m∠4¿ m∠1
m∠4¿ m∠2
1 4

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B. Guided/Practice Activity
Activity 1:
Directions: In this activity, you will compare the sum of the measures of any two
sides of a triangle with the measure of the third side.
1. Cut straws into three pieces, and use the pieces to form a triangle. Measure each
side length to the nearest tenth of a centimeter. In the table below, record the
measures of each side of the triangle from smallest to largest; then, find the sum of
the measures of the small and medium sides. Repeat this activity twice, with two
other triangles, to complete the chart.
Small Medium Large Small + Medium

2. Compare the sum of the measures of the small and medium sides to the
measure of the large side for each triangle you created. Describe what you notice.

3. Cut straws into three pieces so that it is NOT possible to form a triangle. Measure
each side of the non-triangle to the nearest tenth of a centimeter. In the table below,
record the measures of each side of the non-triangle from smallest to largest; then,
find the sum of the measures of the small and medium sides. Repeat this activity
twice, with two other non-triangles, to complete the chart.
Small Medium Large Small + Medium

4. Compare the sum of the measures of the small and medium sides to the measure
of the large side for each non-triangle you created. Describe what you notice.
___________________________________________________________________

5. In the blanks below, write three inequalities that are always true for a triangle with
side lengths s (small side), m (medium side), and l (long side). (These inequalities
should be based on your conclusion from Question 5.)

Triangle Inequality Theorem

In a triangle with side lengths s , m,∧l

______ +______ > ______

______ +______ > ______

______ +______ > ______

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Activity 2.
Directions: Fill up the table by the correct information, and complete the sentence
below.
Segment Increasing Order Inequality Do the 3
Lengths (a, b, c) (a+ b ¿ c) lengths make a
triangle
2, 6 and 1
5, 2 and 3
4, 3 and 6

Short Cut: The ______ of the two ________side must be __________ than
the______________.

Activity 3.
Directions: Use the Triangle Inequality Theorem (a< b+c , b<a+ c , c <a+ b) to solve
for the value of x.
1.

10 x-2

3x

2.

2x x

3x

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C. Independent Activity
Activity 1.
Directions: Complete the table below by following the example in row 1.
Set of Straw Do the Compare the Compare Compare
Pieces straw sum of the (l+m) and k (k+m) and l
forms a lengths of the
triangle shorter straws
or not. (k+ l) with that
of the longest
m.
k l m k+l ¿ ,>, m l+ m ¿ ,>,=¿
k k+m ¿, l
YES NO ¿ ¿,
¿
1. 4 6 9 4+6 ¿ 9 6+9 ¿ 4 4+9 ¿ 4
2. 5 5 10
3. 6 7 11
4. 4 7 10
5. 4 7 12

Activity 2.
Directions: Find the range for the value of x.

1.) 2.)
5-x 2x
x-1 x+2

4 5

3.)
8 x

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Activity 3
Directions: Find the measure of the angle indicated and arrange the sides of the
triangle in descending order.

1. 2. m∠A = (9x – 4)0, m∠B = (4x-16)0

• m∠C = 68-2x
Y
D 15x+5 800 C

6x + 6

D. Evaluation
Directions: Read and understand each question. Write the letter of your correct
answer on the space provided.

_____ 1. Which set of numbers could be the lengths of the sides of a triangle?
A. 6, 9, 15 B. 3, 3, 7 C. 6, 9, 12 D. 1, 2, 3
_____ 2. Which of the following choices CANNOT be the third side length of the
triangle having two side lengths of 15 and 8?
A. 9 B. 13 C. 21 D. 25
_____ 3. How many triangles can you make with these sides. 2, 2, 3, 4, 5?
A. 1 B. 2 C. 3 D. 4
_____ 4. Which of the following combinations could be the sides of triangle?
A. 5, 6, 11 B. 1, 3, 5 C. 5, 16, 20 D. 7, 7, 14
_____ 5. Which of the following represents the range of the third side of the triangle
having two side lengths of 13 and 51?
A. 13 ¿ 3rd side ¿ 51 B. 38¿ 3rd side ¿ 64
C. 38 ≤ 3rd side ≤ 64 D. 3rd side ¿ 64
_____ 6. What could be the measure of the two sides of the triangle having the third
side length ranging 2 6< x <68 ?
A. 21 and 47 B. 23 and 45 C.15 and 18 D. 9 and 59

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_____ 7. What is the least possible measure of the third side if the measure of the
two sides are 4 and 3 ?
A. 1 B. 2 C. 3 D. 4

_____ 8. Three trees are planted. The direct distance from tree A to tree B is 200ft.
The direct distance from tree B to tree C is 300ft, which choices could be the direct
distance from tree A to tree C?
A. 50 ft B. 250 ft C. 450 ft D. 650 ft

_____ 9. Which side of the triangle is the longest side in ∆GAD, with m∠G = 350 and
m∠A = 520 ?
A. GA
´ B. AB
´ C. GB
´ D. NONE

_____ 10. In ∆ABC, AB = 4x+ 2 , BC = 5x – 3 and AC = 2x+4. The perimeter of the

triangle is 36 units. Which angle in the triangle has the largest measure?
A. ∠A B. ∠B C. ∠C D. ∠D

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References:
Abuzo, Emmanuel P., Bryant Merden L., Cabrella Jem Boy B., Caldez, Belen
P., Callanta, Melvin M., Castro, Anastacia Proserfina I., Halabaso, Alicia
R., Javier, Sonia P., Nocom, Roger T., Ternida, Conception S.
(2013).Learners Module Mathematics 8, Department of Education.
Clark, Laura (2004). Discovering the Triangle Inequality Theorem Lesson Plan.
Retrieve on October 14, 2020 at [Online]
https://www.deltastate.edu/docs/math/Ip3Iclark
McFadden, Marianne (2015). A Comprehensive Lesson on the Triangle
Inequality Theorem: Activities and Assessment Method.
Reddick, Erica. Triangle Inequality Investigation Lesson Plan. Retrieve on
October 14, 2020 at [Online]
https://www.cpalms.org/Public/PreviewResourceLesson/Preview/40261
Rowland, Angel (2010). Triangle Inequalities Theorem Lesson Plan. Pierce
County High School Blackshear, GA, US. July 16, 2010. Retrieve on
October 14, 2020 at [Online] http://library.curriki.org/oer/Triangle-
Inequalities-Theorem-Lesson-Plan

Quality Assured/Evaluated by the Following:

Karl Louie B. Panganoron
Ressme M. Bulay-og
Manilyn M. Diadula
Ariel S. Hermoso
Leizl P.tantan

Gina I. Lihao
Education Program Supervisor in Mathematics

Reviewed By:

Evelyn F. Importante
OIC- CID Chief EPS

Raymund M. Salvador
OIC- Assistant Schools Division Superintendent

Jerry C. Bokingkito
OIC- Assistant Schools Division Superintendent

Dr. Jeanelyn A. Aleman, CESE

OIC-Schools Division Superintendent

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