This document provides examples and exercises on using inequalities to solve problems involving triangles. It defines the triangle inequality theorem and shows how to set up and solve inequalities to find possible lengths of triangle sides. Examples demonstrate drawing diagrams to visualize solutions and labeling angles and sides in order of measure. Practice problems have students list triangle sides or angles from least to greatest, determine if given side lengths can form a triangle using inequalities, and find the possible length of a third side of a triangle given the other two sides.

1. 5.5 Use Inequalities in a Triangle
5.5
Bell Thinger
1. Solve 3x + 8 < 29.
ANSWER x < 7
2. Solve 15 > –2x – 9.
ANSWER x > –12
3. Solve 2x – 2 < (3x – 4) + (x – 8).
ANSWER
x>5
4. Find all integer solutions for 4 < x < 11.
ANSWER 5, 6, 7, 8, 9, 10
5. A triangle has angle measures 82° and 34°. Find the
measure of the third angle.
ANSWER
64°

2. 5.5

3. Example 1
5.5
Draw an obtuse scalene triangle. Find the largest
angle and longest side and mark them in red. Find
the smallest angle and shortest side and mark them
in blue. What do you notice?
SOLUTION
The longest side and
largest angle are
opposite each other.
The shortest side and
smallest angle are
opposite each other.

4. Example 2
5.5

5. Example 2
5.5
SOLUTION
Draw a diagram and label the side
lengths. The peak angle is opposite
the longest side so, by Theorem 5.10,
the peak angle is the largest angle.
o
The angle measures sum is 180 , so the third angle
o
measure is 180o – (46o + 59o) = 75 . You can now label
the angle measures in your diagram.
The greatest angle measure is 75o, so the correct
answer is C.

6. Guided Practice
5.5
1. List the sides of RST in
order from shortest to
longest.
ANSWER
ST, RS, TR

7. Guided Practice
5.5
2. Another stage prop is a right triangle with sides
that are 6, 8, and 10 feet long and angles of 90 ,
about 37°, and about 53°. Sketch and label a
diagram with the shortest side on the bottom and
the right angle at the left.
ANSWER

8. 5.5

9. Example 3
5.5
ALGEBRA A triangle has one side of length 12 and
another of length 8. Describe the possible lengths
of the third side.
SOLUTION
Let x represent the length of the third side. Draw
diagrams to help visualize the small and large values
of x. Then use the Triangle Inequality Theorem to write
and solve inequalities.

10. Example 3
5.5
Small values of x
x + 8 > 12
x>4
Large values of x
8 + 12 > x
20 > x,or x < 20
ANSWER
The length of the third side must be greater than
4 and less than 20.

11. Guided Practice
5.5
3. A triangle has one side of 11 inches and another of 15
inches. Describe the possible lengths of the third side.
ANSWER
4 < x < 26

12. Exit Slip
5.5
In Exercises 1 and 2, list the sides or angles in order
from least to greatest.
1.
ANSWER AB, BC, AC

13. Exit Slip
5.5
In Exercises 1 and 2, list the sides or angles in order
from least to greatest.
2.
ANSWER ∠R, ∠T, ∠S

14. Exit Slip
5.5
Tell whether the side lengths can form a triangle.
3. 37 m, 35 m, 18 m
ANSWER yes
4. 3 ft, 3 ft, 6 ft
ANSWER no

15. Exit Slip
5.5
5.
Jeremy wants to build a triangular toy using sticks.
He has one stick that is 12 inches and another that
is 10 inches. What are the possible lengths of the
third side of the triangle?
ANSWER greater than 2 in. and less than 22 in.

16. 5.5
Homework
Pg 345-348
#8, 11, 14, 24, 33