School purposes

1. MATH 8- QUARTER 4
PROVING
INEQUALITIES
IN A
TRIANGLE

2. RECALL TIME!

3. WHAT THEOREMS
CORRESPOND
EACH FIGURE?

4. WHAT THEOREMS
CORRESPONDS EACH
FIGURE?
A. Hinge
Theorem
B. Converse of
Hinge Theorem
C. Triangle
Inequality
Theorem 1
(Ss Aa)
→
D. Triangle
Inequality Theorem
2
(Aa Ss)
→
E. Triangle
Inequality Theorem
3
(S1 + S2 > S3)
F. Exterior Angle
Inequality
Theorem
WORD
BANK

5. WHAT THEOREM
CORRESPONDS THIS
FIGURE?
C. Triangle
Inequality
Theorem 1
(Ss Aa)
→

6. WHAT THEOREM
CORRESPONDS THIS
FIGURE?
WHAT THEOREM
CORRESPONDS THIS
FIGURE?
E. Triangle
Inequality
Theorem 3
(S1 + S2 > S3)

7. WHAT THEOREM
CORRESPONDS THIS
FIGURE?
𝒎∠𝑪𝑶𝑹>𝒎∠𝑶𝑪𝑹
F. Exterior Angle
Inequality
Theorem

8. WHAT THEOREM
CORRESPONDS THIS
FIGURE?
WHAT THEOREM
CORRESPONDS THIS
FIGURE?
A. Hinge
Theorem

9. WHAT THEOREM
CORRESPONDS THIS
FIGURE?
B. Converse of
Hinge Theorem

10. WHAT IS OUR LAST
TOPIC?
APPLYING INEQUALITIES
IN A TRIANGLE

11. MATH 8- QUARTER 4
PROVING
INEQUALITIES
IN A
TRIANGLE

12. At the end of the lesson, learners are expected to:
LESSON OBJECTIVES
State the theorems on triangle inequality,
exterior angle inequality, Hinge and its
converse;
1
Prove statements involving triangle
inequalities using theorems on inequalities in
triangles, and
2
Make a design where triangle inequalities are
applied in real life situation.
3

13. Triangle Inequality Theorem 1 (SsAa)
Given:
Prove:
Proof:
We cannot directly prove that , thus, there is a need to make additional
constructions (see the second figure). Locate S on such that , and connect 𝑡𝑜
with a segment to form a triangle .

14. Triangle Inequality Theorem 1 (SsAa)
STATEMENTS REASONS
1. _______________ (How do you describe
the relationship between and ?)
1. By construction
2. _______________ (Based on statement
1, what kind of a triangle is ∆ ?)
𝑃𝑄𝑆
2. Definition of isosceles
triangle
3. _____________ (Based on statement 1,
how do you describe the relationship
between 1 and 2?)
∠ ∠
3. Base angles of
isosceles triangles are
congruent.
𝑷𝑸 ≅ 𝑷𝑺
∆𝑃𝑄𝑆 is isosceles triangle
∠1 2
≅ ∠
Substitute Property Property of Inequality
Angle Addition Postulate
is isosceles triangle
TWO-COLUMN PROOF:

15. Triangle Inequality Theorem 1 (SsAa)
STATEMENTS REASONS
4. =
𝑚∠𝑄
1+ 3
𝑚∠ 𝑚∠
4. ________________ (What postulate supports the
statement that the sum of the measures of 1 and 3 is
∠ ∠
equal to measure of ?)
∠𝑃𝑄𝑅
5. > 1
𝑚∠𝑄 𝑚∠ 5. _______________ (What property supports the inequality
statement focusing on 1 based on statement 4?)
∠
6. > 2
𝑚∠𝑄 𝑚∠ 6. _______________ (What property supports the step where
the right side of the inequality in statement 5 is replaced
with its equivalent in statement 3?)
Angle Addition Postulate
Property of Inequality
Substitute Property
Substitute Property Property of Inequality
Angle Addition Postulate
is isosceles triangle
TWO-COLUMN PROOF:

16. Triangle Inequality Theorem 1 (SsAa)
STATEMENTS REASONS
7. _____________ (Based on the
illustration, write an operation
statement involving measures of
, , 3.
∠𝑄𝑆𝑅 ∠𝑅 𝑎𝑛𝑑∠
7. The sum of the measures
of the interior angles of a
triangle is 180°
8. 2+ =
𝑚∠ 𝑚∠𝑄𝑆𝑅 ° 8. Linear pair theorem
9. 2+ = +
𝑚∠ 𝑚∠𝑄𝑆𝑅 𝑚∠𝑄𝑆𝑅
+ 3
𝑚∠𝑅 𝑚∠
9. Substitution/ Transitive
Property
𝑚∠𝑄𝑆𝑅 + + 3
𝑚∠𝑃𝑅𝑄 𝑚∠
=180°
Substitute Property Property of Inequality
Angle Addition Postulate
is isosceles triangle
TWO-COLUMN PROOF:

17. Triangle Inequality Theorem 1 (SsAa)
STATEMENTS REASONS
10. _____________ (What will be the result if
is subtracted from both sides of
𝑚∠𝑄𝑆𝑅
statement 9?)
10. Subtraction
Property
11. _____________ (Based on statement number 10,
write an inequality statement focusing on .)
∠𝑅
11. Property of
Inequality
12. ____________ (Based on statements 6 and 11: If
m > 2 and 2 > , then __)
∠𝑃𝑄𝑅 𝑚∠ 𝑚∠ 𝑚∠𝑅
12. Transitive
Property
𝒎∠𝟐=𝒎∠ 𝑷𝑹𝑸+𝒎∠𝟑
𝑚∠2 > 𝑚∠𝑄𝑅𝑃
𝑚∠𝑃𝑄𝑅 > Q
𝑚∠𝑃𝑅
Substitute Property Property of Inequality
Angle Addition Postulate
is isosceles triangle
TWO-COLUMN PROOF:

18. Triangle Inequality Theorem 2 (AaSs)
Given:
Prove:
Proof:
To prove that | |>| |, we will use indirect proof. Thus, we need to
𝑄𝑅 𝑃𝑄
assume that | |≯| |.
𝑄𝑅 𝑃𝑄

19. STATEMENTS REASONS
1. | | = | | | | < |
𝑄𝑅 𝑃𝑄 𝑜𝑟 𝑄𝑅
|
𝑃𝑄
1. Assumption that | | |
𝑄𝑅 ≯
|
𝑃𝑄
2. Consider | | =| |. If |
𝑄𝑅 𝑃𝑄
| = | |, then ∆ is
𝑄𝑅 𝑃𝑄 𝑃𝑄𝑅
an isosceles triangle.
2. Definition of isosceles triangle
3. _____________ (Consequently,
what can you say about and
∠𝑃
?)
∠𝑅
3. Base angles of isosceles
triangles are congruent.
∠ 𝑷 ≅∠ 𝑹
Smaller
Hinge Theorem Triangle Inequality Theorem 1
TWO-COLUMN PROOF:
Triangle Inequality Theorem 2 (AaSs)

20. STATEMENTS REASONS
4. The assumption that | | = | |
𝑄𝑅 𝑃𝑄
is false.
4. The conclusion that ∠ ≅ ∠
𝑃 𝑅
contradicts the given that m >
∠ ∠
𝑃 𝑚 𝑅
5. Consider | | < | |. If | | <
𝑄𝑅 𝑃𝑄 𝑄𝑅
| |, then _________________.
𝑃𝑄
5. ______________ (By what theorem of
triangle inequality?)
6. _____________ (The assumption
that < is true or false?)
𝑄𝑅 𝑃𝑄
6. The conclusion that <
𝑚∠𝑃 𝑚∠𝑅
contradicts the given that m >
∠ ∠
𝑃 𝑚 𝑅
7. _____________ (Therefore, >
𝑄𝑅
must be true or false?)
𝑃𝑄
7. The assumption that | | ≯ | |
𝑄𝑅 𝑃𝑄
contradicts the known fact that m >
∠𝑃
𝑚∠𝑅
𝒎∠𝑷 <𝒎∠ 𝑹
Smaller
Hinge Theorem Triangle Inequality Theorem 1
TWO-COLUMN PROOF:
False
Triangle Inequality Theorem 1
True
Triangle Inequality Theorem 2 (AaSs)

21. Triangle Inequality Theorem 3 (S1+S2S3)
Given:
Prove:
Proof:
Since , then and are true. Hence, what remains to be proven is the last
statement, .

22. STATEMENTS REASONS
1. _______________ (How do you describe
the relationship between and ?)
1. By construction
2. _______________ (Based on statement
1, what kind of a triangle is ∆ ?)
𝑃𝑄𝑆
2. Definition of isosceles
triangle
3. _____________ (Based on statement 1,
what is the relationship between
and ?)
∠𝑃𝑅𝑆 ∠𝑃𝑆𝑅
3. Base angles of isosceles
triangles are congruent.
∠ 𝑷𝑹𝑺 ≅∠ 𝑷𝑺𝑹
Transitive Property
Substitution Property ∆𝑃𝑅𝑆 is equilateral triangle
∆𝑃𝑅𝑆 is isosceles triangle ∆𝑃𝑅𝑆 is scalene triangle
Triangle Inequality Theorem 1 (Ss→ Aa) Reflexive Property of Equality Triangle Inequality Theorem 2 (Aa→Ss)
TWO-COLUMN PROOF:
is isosceles triangle
𝑷𝑺 ≅ 𝑷𝑹
Triangle Inequality Theorem 3 (S1+S2S3)

23. ∠ 𝑷𝑹𝑺 ≅∠ 𝑷𝑺𝑹
STATEMENTS REASONS
4. Based on the illustration,
∠𝑃𝑆𝑅 ≅∠ 𝑄𝑆𝑅
4. ______________ (By what
property of equality?)
5. If (statement 3)
∠𝑃𝑅𝑆≅∠𝑃𝑆𝑅
and (statement 4),
∠𝑃𝑆𝑅≅ ∠𝑄𝑆𝑅
then ________________.
5. Transitive Property of
Equality
6. From the illustration, m =
∠𝑄𝑅𝑆
+
𝑚∠𝑃𝑅𝑄 𝑚∠𝑃𝑅𝑆
6. _____________ (By what
postulate?)
Transitive Property
Substitution Property ∆𝑃𝑅𝑆 is equilateral triangle
∆𝑃𝑅𝑆 is isosceles triangle ∆𝑃𝑅𝑆 is scalene triangle
Triangle Inequality Theorem 1 (Ss→ Aa) Reflexive Property of Equality Triangle Inequality Theorem 2 (Aa→Ss)
TWO-COLUMN PROOF:
∠𝑷𝑹𝑺≅∠𝑸𝑺𝑹
Reflexive Property of
Equality
Angle Addition Postulate
Triangle Inequality Theorem 3 (S1+S2S3)

24. ∠ 𝑷𝑹𝑺 ≅∠ 𝑷𝑺𝑹
Transitive Property
Substitution Property ∆𝑃𝑅𝑆 is equilateral triangle
∆𝑃𝑅𝑆 is isosceles triangle ∆𝑃𝑅𝑆 is scalene triangle
Triangle Inequality Theorem 1 (Ss→ Aa) Reflexive Property of Equality Triangle Inequality Theorem 2 (Aa→Ss)
TWO-COLUMN PROOF:
STATEMENTS REASONS
7. Based on statements 5 and 6,
= +
𝑚∠𝑄𝑅𝑆 𝑚∠𝑃𝑅𝑄 𝑚∠𝑄𝑆𝑅
7. _____________ (By what
property?)
8. Based on statement 7,
>
𝑚∠𝑄𝑅𝑆 𝑚∠𝑄𝑆𝑅
8. Property of Inequality
9. | |>| |
𝑄𝑆 𝑄𝑅 9. ______________ (By what
theorem of triangle
inequality?)
Substitution Property
Triangle Inequality
Theorem 2 (Aa→Ss)
Triangle Inequality Theorem 3 (S1+S2S3)

25. Transitive Property
Substitution Property ∆𝑃𝑅𝑆 is equilateral triangle
∆𝑃𝑅𝑆 is isosceles triangle ∆𝑃𝑅𝑆 is scalene triangle
Triangle Inequality Theorem 1 (Ss→ Aa) Reflexive Property of Equality Triangle Inequality Theorem 2 (Aa→Ss)
TWO-COLUMN PROOF:
STATEMENTS REASONS
10. 10. Segment Addition
Postulate
11._____________ (Write a statement
using statements 9 and 10.)
11. Substitution Property
12.____________ (Write a statement
using statements 1 and 11.)
12. Substitution Property
𝑷𝑸 + 𝑷𝑺 >𝑸𝑹
¿ 𝑷𝑸∨+¿ 𝑷𝑹∨¿∨𝑸𝑹∨¿
Triangle Inequality Theorem 3 (S1+S2S3)

26. Exterior Angle Inequality Theorem
Given: with exterior angle
∆𝑃𝑄𝑅 ∠𝑃𝑅𝑆
Prove:
Proof:
Let us prove that > by constructing the following:
𝑚∠𝑃𝑅𝑆 𝑚∠𝑄𝑃𝑅
 midpoint on such that
𝑇 ≅
 through T such that ≅

27. STATEMENTS REASONS
1. ; 1. By construction
2. _____________ (What is the
relationship between 3 and 4?)
∠ ∠
2. Vertical Angles are
congruent
3. _____________ (Based on statements 1
and 2, relate two triangles in the
illustration)
3. SAS Triangle
Congruence Postulate
∆ 𝑷𝑻𝑸 ≅ ∆ 𝑹𝑻𝑼
TWO-COLUMN PROOF:
∠𝟑≅∠𝟒
Exterior Angle Inequality Theorem

28. STATEMENTS REASONS
4. _______________.
(Based on question number 2)
4. Corresponding parts of congruent
triangles are congruent
5. = ______________.
𝑚∠𝑃𝑅𝑆
(Based on the illustration)
5. Angle Addition Postulate
6. 6. Property of Inequality
7. ______________ (Using
statement in question 3 and
7. Substitution Property
𝒎 ∠ 𝑷𝑹𝑺>𝒎∠ 𝑸𝑷𝑹
TWO-COLUMN PROOF:
∠𝑸𝑷𝑹≅∠𝟏
𝒎 ∠𝟏+𝒎 ∠𝟐
Exterior Angle Inequality Theorem

29. Hinge Theorem or SAS Inequality Theorem
Given: ,
Prove:
TWO-COLUMN PROOF:
STATEMENTS REASONS
1. 1.
2. 2.
3. 3.
4. 4.
𝐴𝐹 ≅ 𝑀𝑅
𝑚∠𝐴𝑅𝐹>𝑚∠ 𝐴𝑅𝑀
𝐴𝑅 ≅ 𝐴𝑅
𝐴 𝑅 > 𝐴 𝑀
Given
Given
Reflexive Property
Hinge Theorem or SAS
Inequality Theorem

30. Converse of Hinge Theorem or SSS
Inequality Theorem
Given:
Prove:
TWO-COLUMN PROOF:
STATEMENTS REASONS
1. 1.
2. 2.
3. 3.
𝐴 𝑀 ≅ 𝐻𝑇
𝑇𝑀 ≅ 𝑇𝑀
𝐴 𝑇 > 𝐻 𝑀
Given
Reflexive Property
ConverseofHingeTheoremorSSSInequalityTheorem

31. NOW IT’S
YOUR TURN!
You’ve just learned how to find a
number’s factors. Time to keep
practicing with today’s assignment.

32. ASSIGNMENT
Directions: Observe your surroundings and make a
design where triangle inequalities are applied. Make
justifications that the triangle inequality exists in the
created design.

33. ASSIGNMENT: Rubric

34. RESOURCES