1. Use Inequalities in A Triangle

2. Objectives:
Use triangle measurements to decide which side is longest or which angle is largest.
Use the Triangle Inequality Theorem

3. Comparing Measurements of a 
shortest side
smallest angle
longest side
largest angle
The longest side and largest angle of a  are opposite each other.
The shortest side and smallest angle of a  are opposite each other.

4. 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.
mA > mC

5. 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.
60°
40°
EF > DF

6. Example 1: Writing Measurements in Order from Least to Greatest
Write the measurements of the triangles from least to greatest.
m G < mH < m J
JH < JG < GH
100°
45°
35°

7. 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 8 5 7

8. 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.

9. Activity: Constructing a Triangle
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.
Blue straw: 2cm
Green straw: 3 cm
Red straw: 4cm
Yellow straw: 5cm

10. Activity: Constructing a Triangle
2 cm, 2 cm, 5 cm
3 cm, 2 cm, 5 cm
4 cm, 2 cm, 5 cm
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.

11. 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

12. Example 3: Finding Possible Side Lengths
A triangle has one side of 10 cm and another of 14 cm. Describe the possible lengths of the third side
SOLUTION: Let x represent the length of the third side. Using the Triangle Inequality, you can write and solve inequalities.
If x was the smallest side, then x + 10 > 14, so x > 4 If x was the longest side, then > x, so 24 > x ► So, the length of the third side must be greater than 4 cm and less than 24 cm.

13. Example 4: Using Algebra to Find Possible Side Lengths
Solve the inequality:
AB + AC > BC.
(x + 2) +(x + 3) > 3x – 2
2x + 5 > 3x – 2
5 > x – 2
7 > x

14. Assignment
Handout on Triangle Inequalities