5-5Triangle Inequality

You recognized and applied properties of inequalities to the relationships between the angles and sides of a triangle.

• Use the Triangle Inequality Theorem to identify possible triangles.

• Prove triangle relationships using the Triangle Inequality Theorem.

Is It a Triangle?1. Break one piece of linguini into three

different length pieces.

2. Use your three pieces to make a triangle. Set aside.

3. Repeat the process with your other piece of linguini. Make a triangle.

4. Put two pieces of your triangle end to end. Is it longer than the third piece?

Page 364

Can these lengths be the sides of a triangle?

1. 4, 7, 10

2. 3, 5, 8

3. 3, 5, 7.9

1.Yes

2.No

3.Yes

Is it possible to form a triangle with side lengths of 6.8, 7.2, 5.1? If not, explain why not.

Answer: yes

Check each inequality.

6.8 + 7.2 > 5.1 7.2 + 5.1 > 6.8 6.8 + 5.1 > 7.2

14 > 5.1 12.3 > 6.8 11.9 > 7.2

Since the sum of all pairs of side lengths are greater than the third side length, sides with lengths 6.8, 7.2, and 5.1 will form a triangle.

A. yes

B. no

Two side of a triangle measure 6 cm and 9 cm. Write an inequality that represents the range of values for the possible lengths of the third side.

6

9?

6 9

?

x + 6 > 9

x > 3

6 + 9 > x

15 > x3 < x < 15

A. 4

B. 9

C. 12

D. 16

In ΔXYZ, XY = 6, and YZ = 9. Which measure cannot be XZ?

TRAVEL The towns of Jefferson, Kingston, and Newbury are shown in the map below. Prove that driving first from Jefferson to Kingston and then Kingston to Newbury is a greater distance than driving from Jefferson to Newbury.

Answer: By the Triangle Inequality Theorem, JK + KN > JN. Therefore, driving from Jefferson to Kingston and then Kingston to Newbury is a greater distance than driving from Jefferson to Newbury.

Abbreviating the vertices as J, K, and N: JK represents the distance from Jefferson to Kingstown; KN represents the distance from Kingston to Newbury; and JN the distance from Jefferson to Newbury.

What has to be true about the length of the sides of a triangle?

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

5-5 Assignment

Page 367, 6-17