The document discusses several theorems and properties related to inequalities in one and two triangles: - The triangle inequality theorem states that the length of any side of a triangle is less than the sum of the lengths of the other two sides. - A related theorem is that if two sides of a triangle are congruent to the sides of another triangle but the included angles are not equal, then the third sides will be unequal, with the longer side opposite the larger angle. - Properties are also discussed regarding the relationship between side lengths and angle measures, such that the longer side is always opposite the larger angle.

1. Inequalities in One Triangle
and in Two Triangles

2. LOOP-A-WORD
Find the listed words in the
puzzle. They are arranged
horizontally, vertically,
diagonally, upward or
backward.

4. LOOP-A-WORD
Interior
Theorem
Inequality
Exterior
Remote
Triangle
Adjacent
Hinge
Angle
SAS

5. f
f
f

6. MATH-ionary
Define all the
words in the said
Word Hunt.

7. MATH-ionary
❑ Exterior – Angle outside the triangle
❑Hinge- If two sides of a triangle are congruent to the
sides of another triangle, but their included angles
are not, then the remaining sides are unequal. The
longer side is opposite the larger angle.
❑Angle- part of the triangle
❑SAS – Side-Angle-Side (with included angle)

8. MATH-ionary
❑Inequality- not equal
❑Theorem- statement that must be proven
❑Interior- Angles inside the triangle
❑Triangle- closed figure which has three sides
❑Remote- angles of a triangle that are not adjacent to
a given angle.
❑Adjacent- angles having a common vertex and a
common side.

9. Observe the figure
C
6 3
B 2
4
1 5
A

10. Question
1. What are the exterior angles
of the given triangle?
2. How will you describe the
exterior angle?

11. Exterior Angle of a Triangle
An exterior angle of a
triangle is an angle that forms
a linear pair with an interior
angle of a triangle is extended.

12. Observe the figure
A
B C
4 3
5

13. 1. What is the sum of the lengths of
AB and AC ?
2. Compare the sum of the lengths of
AB and AC to BC
3. What is the sum of the lengths of
AB and BC?
Question

14. 4. Compare the sum of the
lengths of AB and BC to
AB
5. What is the sum of the
lengths of AC and BC?
Question

15. 6. Compare the sum of the lengths
of AC and BC to AB.
7. What can you say about the
sum of the two sides of a triangle
compared to its third side?
Question

16. Triangle Inequality
The length of a side of a triangle is
less than the sum of the lengths of the
other two sides. The length of one side
is also greater than the positive
difference of the lengths of the other
two sides.

17. Observe the figure
100°
A C
7 4
10
25° 55°
B

18. 1. What is the largest angle?
2. What is the smallest angle?
3. What is the length of the side
opposite the largest angle?
Question

19. 4. What is the length of the side
opposite the smallest angle?
5. What is the relationship
between the length of the side and
the angle opposite to the given
side?
Question

20. Theorem
If the length of the two sides of a
triangle is unequal, the measures
of the angles are also unequal.
The longer side is opposite the
angle with a greater measure.

21. Triangle Inequality Theorem 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. In other
words, opposite the largest angle is the
longest side.

22. Observe the figure
97°
A C
9 7
15
20° 63°
B

23. 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. In symbol,
a+ b > c; a + c > b; b+ c > a

24. Observe the figure
97°
A C
9 7
15
20° 63°
B

25. Observe the figure
A
B
D C
E
F
56° 50°
9 9
10 10

26. 1. What are the congruent
sides?
2. What have you notice on
the included angles of the
two triangles?
Question

27. Question
What conclusion can you make
about the opposite sides of the
included angles of the two
triangles? Are they congruent?
Which side is longer?

28. Hinge Theorem
If two sides of a triangle are
congruent to the sides of another
triangle, but their included angles are
not, then the remaining sides are
unequal. The longer side is opposite
the larger angle.

29. Things to Remember!
Exterior Angle of a Triangle- An
exterior angle of a triangle is an
angle that forms a linear pair
with an interior angle of a
triangle is extended.

30. Things to Remember!
Triangle Inequality Postulate- The length
of a side of a triangle is less than the sum
of the lengths of the other two sides. The
length of one side is also greater than the
positive difference of the lengths of the
other two sides.

31. Things to Remember!
Theorem- If the length of the two
sides of a triangle is unequal, the
measures of the angles are also
unequal. The longer side is opposite
the angle with a greater measure.

32. Things to Remember!
Hinge Theorem- If two sides of a
triangle are congruent to the sides of
another triangle, but their included angles
are not, then the remaining sides are
unequal. The longer side is opposite the
larger angle.

33. LET ’S DO THIS!
Fill in the blanks with the
correct relation symbol
( >, < ) to show the
relationship.

34. D
25
56
40
m ∠O
m ∠G
m ∠G
m ∠O
1. m ∠D
2. m ∠D
3. m ∠O
4. m ∠G
5. m ∠O m ∠D
O G

35. B K
65°
O
55°
60°

36. LET ’S DO THIS!
Is it possible for a triangle to have
sides with the length indicated?
1. 3, 4, 5
2. 8, 7, 10
3. 2, 5, 6

37. F
U
N
1
2
3
1. m ∠2
2. m ∠1
3. m ∠4
m ∠4
m ∠4
m ∠2
4

38. SEATWORK
Name the largest and
the smallest angle of
the triangle.

39. R
T
S
22
24 23
Largest Angle- ∠S
Smallest Angle- ∠T

40. W
18
U
16
17
V
Largest Angle- ∠V
Smallest Angle- ∠U

41. A
N
75
°
D
70°
35
°

42. A
T
F
87
°
43°
50°

43. T
H
123
°
32°
G
25°