5.5 Use Inequalities in A Triangle
Objectives:
 Use triangle measurements to decide which
side is longest or which angle is largest.
 Use the Triangle Inequality Theorem
Comparing Measurements of a 
largest angle
 The longest side and largest
angle of a  are opposite
each other.
longest side
 The shortest side and
smallest angle of a  are
opposite each other.
shortest side
smallest angle
Theorem 5.10
If one SIDE of a
triangle is longer than
another SIDE, then the
ANGLE opposite the
longer side is larger
than the ANGLE
opposite the shorter
side.
B
3
5
A
C
mA
> mC
Theorem 5.11
If one ANGLE of a
triangle is larger than
another ANGLE, then
the SIDE opposite the
larger angle is longer
than the SIDE opposite
the smaller angle.
D
60°
40°
F
EF
> DF
E
Example 1:
Writing Measurements in Order from Least to Greatest
Write the
measurements of the
triangles from least to
greatest.
J
100°
45°
m G < mH < m J
JH < JG < GH
H
35°
G
Example 2:
Writing Measurements in Order from Least to Greatest
Write the
measurements of the
triangles from least to
greatest.
QP < PR < QR
m R < mQ < m P
R
8
Q
7
5
P
Using the Triangle Inequality
 Not every group of three segments can be used to
form a triangle. The lengths of the segments must fit
a certain relationship.
Activity: Constructing a Triangle
a.
b.
c.
2 cm, 2 cm, 5 cm
3 cm, 2 cm, 5 cm
4 cm, 2 cm, 5 cm
Activity:
Let’s try drawing triangles with the given side
lengths.
Activity: Constructing a Triangle
a.
b.
c.
2 cm, 2 cm, 5 cm
3 cm, 2 cm, 5 cm
4 cm, 2 cm, 5 cm
2
2
5
C
D
D
3
4
2
A
5
2
B
A
5
Notice, only group (c) is possible. Thus, what we can deduce is
that the sum of the first and second lengths must be greater than
the third length.
B
Theorem 5.12: Triangle Inequality Theorem
 The sum of the lengths
of any two sides of a
triangle is greater than
the length of the third
side.
AB + BC > AC
AC + BC > AB
AB + AC > BC
A
C
B
Example 3:
Finding Possible Side Lengths
 A triangle has one side of
If x was the smallest side,
then
x + 10 > 14, so
x>4
 SOLUTION: Let x
If x was the longest side,
then
10 + 14 > x, so
24 > x
10 cm and another of 14
cm. Describe the
possible lengths of the
third side
represent the length of
the third side. Using the
Triangle Inequality, you
can write and solve
inequalities.
► So, the length of the
third side must be
greater than 4 cm and
less than 24 cm.
Example 4:
Using Algebra to Find Possible Side Lengths
 Solve the inequality:
A
AB + AC > BC.
x+ 2
B
x+ 3
3x - 2
C
(x + 2) +(x + 3) > 3x – 2
2x + 5 > 3x – 2
5>x–2
7>x
Assignment
 Workbooks Pg. 97 – 99 #4 – 9, 13 – 29