5.5 Use Inequalities in a Triangle Goal  Find possible side lengths of a triangle.
Example: Relate side length and angle measure
Mark the largest angle, longest side, smallest angle, and shortest side
of the triangle shown at the right. What do you notice?
Solution
The longest side and largest angle are _____
from each other.
The shortest side and smallest angle are _____
from each other.
THESE ARE SUMMARIZED IN THESE TWO THEOREMS:
THEOREM:
AB > BC, So m  __ > m  ___.
If one side of a triangle is longer than another side,
then the angle opposite the longer side is _______ than the angle opposite the shorter side.
THEOREM:
m  A > m  C, so ___ >____.
If one angle of a triangle is larger than another angle,
then the side opposite the larger angle is ______ than the side opposite the smaller angle.
Example .
_____ < _____ < _____
m____ < m____ < m____
Example.
Name the shortest & longest
side of the triangle.
_____ < _____ < _____
m____ < m____ < m____
Example.
Name the smallest & largest
angle of the triangle.
Checkpoint Complete the following exercise.
List the sides of ∆PQR in order from shortest to longest.
THEOREM: TRIANGLE INEQUALITY THEOREM
The sum of the lengths of any two sides
of a triangle is greater than the length of the third side.
____+ _____ > AC
AC + ___ > ____
____ + AC > _____
Example Find possible side lengths
A triangle has one side of length 14 and another of length 10. Describe the possible lengths of the third side.
Solution It’s helpful to draw a picture:
Describe the possible values of x.
 Write & solve 3 inequalities:
 Write & solve 3 inequalities:
Example Decide if it’s possible to construct a triangle with the given side lengths.
Ask yourself: When I add every possible combination of two sides together is the SUM always
bigger than the remaining side?
a.
b.