1. Class Greeting

2. Objective: The students will apply the Triangle Inequality Theorem.

3. A triangle is formed by three segments, but not every set of three segments can form a triangle.

4. A certain relationship must exist among the lengths of three segments in order for them to form a triangle.

5. Example 3A: Applying the Triangle Inequality Theorem
Tell whether a triangle can have sides with the given lengths. Explain.
7, 10, 19
No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.

6. Example 3B: Applying the Triangle Inequality Theorem
Tell whether a triangle can have sides with the given lengths. Explain.
2.3, 3.1, 4.6   
Yes—the sum of each pair of lengths is greater than the third length.

7. Example 3C: Applying the Triangle Inequality Theorem
Tell whether a triangle can have sides with the given lengths. Explain.
n + 6, n2 – 1, 3n, when n = 4.
Step 1 Evaluate each expression when n = 4.
n + 6
n2 – 1 3n
4 + 6
(4)2 – 1
3(4) 10 15 12

8. Example 3C Continued
Step 2 Compare the lengths.   
Yes—the sum of each pair of lengths is greater than the third length.

9. Check It Out! Example 3a
Tell whether a triangle can have sides with the given lengths. Explain.
8, 13, 21
No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.

10. Check It Out! Example 3b
Tell whether a triangle can have sides with the given lengths. Explain.
6.2, 7, 9   
Yes—the sum of each pair of lengths is greater than the third side.

11. Check It Out! Example 3c
Tell whether a triangle can have sides with the given lengths. Explain.
t – 2, 4t, t2 + 1, when t = 4
Step 1 Evaluate each expression when t = 4.
t – 2 4t
t2 + 1
4 – 2
4(4)
(4)2 + 1 2 16 17

12. Check It Out! Example 3c Continued
Step 2 Compare the lengths.   
Yes—the sum of each pair of lengths is greater than the third length.

13. Example 4: Finding Side Lengths
The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side.
Let x represent the length of the third side. Then apply the Triangle Inequality Theorem.
x + 8 > 13
x + 13 > 8
> x
x > 5
x > –5
21 > x
Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches.

14. Check It Out! Example 4
The lengths of two sides of a triangle are 22 inches and 17 inches. Find the range of possible lengths for the third side.
Let x represent the length of the third side. Then apply the Triangle Inequality Theorem.
x + 22 > 17
x + 17 > 22
> x
x > –5
x > 5
39 > x
Combine the inequalities. So 5 < x < 39. The length of the third side is greater than 5 inches and less than 39 inches.

15. Example 5: Travel Application
The figure shows the approximate distances between cities in California. What is the range of distances from San Francisco to Oakland?
Let x be the distance from San Francisco to Oakland.
x + 46 > 51
x + 51 > 46
> x
x > 5
x > –5
97 > x
5 < x < 97
The distance from San Francisco to Oakland is greater than 5 miles and less than 97 miles.

16. Check It Out! Example 5
The distance from San Marcos to Johnson City is 50 miles, and the distance from Seguin to San Marcos is 22 miles. What is the range of distances from Seguin to Johnson City?
Let x be the distance from Seguin to Johnson City.
x + 22 > 50
x + 50 > 22
> x
x > 28
x > –28
72 > x
28 < x < 72
The distance from Seguin to Johnson City is greater than 28 miles and less than 72 miles.

17. Kahoot!

18. Lesson Summary:
Objective: The students will apply the Triangle Inequality Theorem.

19. Preview of the Next Lesson:
Objective: The students will solve problems using Inequalities in Two Triangles.

20. Stand Up Please