5.5 Triangle Inequalities
Objectives:
1. To complete and
use the Triangle
Inequality Theorems
Assignment:
• P. 331-334: 1, 2, 6-12
even, 16-20, 22-26
even, 27, 30, 34, 35,
36, 46, 49
• Challenge Problems
Objective 1
You will be able
to complete and use
the Triangle Inequality
Theorems
Example 1
On a number line, graph the following
inequalities:
1. 𝑥 > −5
2. −5 ≤ 𝑥 ≤ 5
3. 𝑥 < −5 or 𝑥 > 5
Example 2
Use a whiteboard
to graph the
inequality
𝑦 < 2𝑥 − 5.
Definition of Inequality
For any real numbers 𝑎 and 𝑏, 𝑎 > 𝑏 if
and only if there is a positive number 𝑐
such that 𝑎 = 𝑏 + 𝑐.
10 > 6 because 10 = 6 + 4
If 𝑚∠1 + 𝑚∠2 = 𝑚∠3, then 𝑚∠1 < 𝑚∠3
Example 3
Given three segments of any length, can you
construct a triangle?
Investigation 1
Use the following
investigation to
complete the Triangle
Inequality Theorem.
Oh, and don’t lick the
envelopes. Or eat,
squash, or mix up the
straws. Thanks.
Investigation 1
1. Assemble a triangle
with each set of straws.
You are not allowed to
cut, bend, or otherwise
change the size or
shape of each straw.
2. Were you able to
construct a triangle
each time? Why or
why not?
Investigation 2
1. Try to construct a triangle using segments 1, 3,
and 4.
2. Try to construct a triangle using segments 2, 5,
and 6.
Investigation 2
3. Experiment with other combinations of
segments. Name a set of segments that forms
a triangle and a set of segments that do not
form a triangle.
Triangle Inequality
So this is what happens
when two sides of a
“triangle” together are
smaller than the third
side:
And here’s what
happens with two
sides of a “triangle”
together are equal to
the third side:
Triangle Inequality Theorem
The sum of the
lengths of any
two sides of a
triangle is
greater than
the length of
the third side.
Example 4
Determine whether it is possible to draw a
triangle with sides of the given measures.
1. 1 cm, 2 cm, 3 cm
2. 21 in, 32 in, 18 in
3. 11 m, 6 m, 2 m
Example 5
The two measures of two sides of a triangle
are given. Between what two numbers
must the measure of the third side fall?
Write your answer as a compound
inequality.
1. 21 and 27
2. 5 and 11
3. 30 and 30
Example 6
Find all possible values of x.
Investigation 3
Use the following
Investigation to
discover the
relationship
between the
measures of angles
in triangles and the
lengths of the sides
opposite them.
Investigation 3
1. Draw a large
scalene triangle.
Some group
members should
draw acute
triangles, and
some should
draw obtuse
triangles.
Investigation 3
2. Measure the
angles in each
triangle. Label
the angle with
greatest measure
∠𝐿, the angle with
second greatest
measure ∠𝑀, and
the remaining
angle ∠𝑆.
𝑀
𝑆
𝑀
𝐿
𝐿
𝑆
Investigation 3
3.
Measure the three
sides. Which side is
the longest? Label it
by placing the
lowercase letter l near
the middle of the side.
Which side is the
second longest? Label
it m in the same way.
Which side is the
shortest? Label it s.
𝑀
𝑚
𝑆
𝑀
𝑠
𝑙
𝐿
𝑚
𝐿
𝑙
𝑠
𝑆
Investigation 3
3.
Measure the three
sides. Which side is
the longest? Label it
by placing the
lowercase letter l near
the middle of the side.
Which side is the
second longest? Label
it m in the same way.
Which side is the
shortest? Label it s.
𝑀
𝑚
𝑆
𝑀
𝑠
𝑙
𝐿
𝑚
𝐿
𝑙
𝑠
𝑆
Investigation 3
Which side, l, m, or s,
is opposite the
angle with the
greatest measure?
Which side is
opposite the angle
with the least
measure?
Investigation 3
Which side, l, m, or s,
is opposite the
angle with the
greatest measure?
Which side is
opposite the angle
with the least
measure?
Side-Angle Inequality Theorem
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.
Angle-Side Inequality Theorem
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.
Example 7
In the triangle at the
right, put the
unknown measures
in order from
greatest to least.
55
b
c
68
a
Example 8
Prove the Side-Angle Inequality Theorem.
A
D
1
2
3
C
B
Example 8
Prove the Side-Angle Inequality Theorem.
A
D
1
2
3
C
B
5.5 Triangle Inequalities
Objectives:
1. To complete and
use the Triangle
Inequality and SideAngle Inequality
Theorems
Assignment:
• P. 331-334: 1, 2, 6-12
even, 16-20, 22-26
even, 27, 30, 34, 35,
36, 46, 49
• Challenge Problems