TRIANGLE INEQUALITY THEOREM

Inequalities in One Triangle
and in Two Triangles

Reaching Goals
❑ State the theorem on triangle inequalities,
exterior angle inequality, and hinge theorem.
❑Illustrate the theorem on triangle inequalities,
exterior angle inequality, and hinge theorem.
❑Develop positive attitude towards work.

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

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

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

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)

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.

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

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

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.

Observe the figure
A
B C
4 3
5

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

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

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

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.

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

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

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

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.

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.

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

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 or
b + c > a

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

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

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.

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

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

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

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

Group Activity
Name the largest and
the smallest angle of
the triangle.

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

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

Group Activity
Name the longest side
and the shortest side of
the triangle. .

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.

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.

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.

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.

TRIANGLE INEQUALITY THEOREM

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