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

The longest side and largest angle of a are opposite each other.

The shortest side and smallest angle of a are opposite each other.

shortest side

smallest angle

longest side

largest 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.

53

A

B

C

mA > mC

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.

EF > DF

DE

F

60° 40°

Example 1: Writing Measurements in Order from Least to Greatest

Write the measurements of the triangles from least to greatest.

m G < mH < m J

JH < JG < GH

H

J

G

45°

100°

35°

Write the measurements of the triangles from least to greatest.

QP < PR < QR

m R < mQ < m P

Q

R

P

8

5 7

Example 2: Writing Measurements in Order from Least to Greatest

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. 2 cm, 2 cm, 5 cmb. 3 cm, 2 cm, 5 cmc. 4 cm, 2 cm, 5 cm

Activity: Let’s try drawing triangles with the given side lengths.

Blue straw: 2cm Green straw: 3 cmRed straw: 4cmYellow straw: 5cm

a. 2 cm, 2 cm, 5 cm

b. 3 cm, 2 cm, 5 cm

c. 4 cm, 2 cm, 5 cm

5

22

5

2

3

A B

CD

5

2

4

A B

D

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.

Activity: Constructing a Triangle

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

C

A

B

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, sox > 4

If x was the longest side, then

10 + 14 > x, so24 > x

► So, the length of the third side must be greater than 4 cm and less than 24 cm.

Example 3: Finding Possible Side Lengths

3x - 2

x + 3x + 2

A

B C

Solve the inequality:

AB + AC > BC.

(x + 2) +(x + 3) > 3x – 2

2x + 5 > 3x – 2 5 > x – 2 7 > x

Example 4: Using Algebra to Find Possible Side Lengths

AssignmentHandout on Triangle Inequalities